This excludes metal stakes, knives etc. to the heart, it's not the bleeding out that's the issue. Just for some reason, a wooden stake to the heart ends a vampire. Why?
Assume the type of vampire that is common in fiction (can't go out during the day, doesn't like garlic or crosses, but can pass as human), no Twilight sparkle people please. Vampirism in the context of this post is spread via a transmittable virus that lives in saliva and blood.
Now, a few answerers have claimed that the stake need not be wooden. While this may be true in the traditional vampiric lore, we exist only to answer the specific question here.
Why Wood
Vampire Blood contains a specific set of proteins and nutrients that allow for almost instantaneous regeneration. These proteins are denatured by heat (fire), and cannot re-attach a separated limb (head). As metal and stone stakes are not organic, they could not possibly prevent the regeneration factor from doing its work. Bone also is not effective, as the regeneration factor cannot penetrate the cortical layer on the outside of bones. Wood is therefore the only readily available organic stake medieval peasants would have access to. When the wood is stabbed into the heart, tiny particles break off and are quickly spread throughout the bloodstream. The regeneration factor acts on these particles, causing thousands of small blood clots and massive internal bleeding. If further damage was necessary to kill, blood clots will soon reach the brain, causing massive strokes, rendering the vampire unable to fight back.
Wooden stakes to the arm are not as effective, as the particulate does not spread as far and as quickly, allowing the vampiric immune system to react and destroy the particulate.
Now, you could make a stake out of compressed and baked kelp, or algae, or some other form of organic matter, but who in the world would think of doing that? The only way we found out that wood does the job is through sheer chance. As very, very few people carry around farming implements, weapons, or tools made of non-wood, non-bone organic matter, it would take an enormous amount of time for such a kill to occur by accident.
A kill with Wood would (lol) only require that a farmer's hoe, or a labourer's shovel, or a soldier's spear to break and be used in a desperate attempt to save one's own life.
It's due to the nature of vampires, and their regenerative state. If a metal or other non-organic substance is thrust into their body then over time their regenerative abilities will push it out. Even if they are completely impaled the stake will gradually migrate as the body moves to one side of the coffin or the other.
Use a material that was once alive though (wood is most common but bone would also work) and the regeneration extends to that material and causes it to fuse with the body and trap the vampire in a disabled state. Wood is ideal as over time it actually sprouts roots that spread through the vampires body and immobilize it permanently.
Regardless of what Hollywood might tell you, the stake doesn't have to be wooden, nor does it actually kill the vampire.
The tradition is that to prevent a vampire from rising they need to be staked into the coffin, metal stakes are acceptable, but wood is cheaper and more readily available to the average superstitious peasant. Also reference can be made to the true cross being wood. Removing the stake allows the vampire to rise again.
If you want to kill a vampire you need to cut off its head (stuffing with garlic optional), burn it, or expose it to sunlight.
Any scientific reason would have to be in the make up of wood itself: See The Chemistry of Wood.
Perhaps because wood was living cells; something the cells produce; everything from sugars (like in sap) to cell walls -- or a "recipe" of such products in a particular balance. Think of a chemical recipe as producing a kind of molecular machine, and a machine that won't work if any element is too much or too little: Just like a cake won't rise correctly if it has too much or too little liquid, flour, baking powder, salt, sugar, etc.
Or the science can be in the actual DNA of the wood: DNA and genes are physical things, and the DNA unique to woody plants could be the machine that is causing the vampire death.
In this case for vampire lethality, "wood" must have a precise combination of cellulose, lignins, and other atoms in precise position and combination to cause an interaction with the vampire heart that degrades it and poisons the rest of the vampire system.
If you want the vampire to be dusted, "Scientifically" we'd say whatever biological compound was holding the vampire cells together was defeated by the wood, and the unraveling of each cell causes neighboring cells to unravel: thus a massive chain reaction that unravels every cell of the vampire, effectively turning it to dust. Such chain reactions are not unusual at all in chemistry.
Why the Heart? Well, the heart is obviously specialized cells that do not occur anywhere else in the body; so it has something to do with the nature of those cells. Perhaps they are uniquely vulnerable to the effect of the wood, the only cells in the body with the particular atomic configuration that the wood "fits" on the outside of the cell; so just touching wood gets the wood "recipe" inside the heart cell; where the cell replicates it (like a virus) to the point of rupture: then the physical force of that rupture infects other cells, which rupture, and voila! exponential growth of a disintegrating core; a chain reaction.
Of course if there is science, there is a chance of isolating the compound in wood (by experimenting on vampires, or vampire heart tissue), so something like steel stakes or bullets could be coated in it; which might make for easier delivery. But it still must come into direct contact with a (formerly) beating heart cell, so must penetrate the chest. But something like bullets might be needed if a vampire had (through surgery) encased his heart in a hardened steel shell.
Vampire hearts typically do not beat; so they don't circulate blood. So a wooden stake elsewhere in the vampire body touches no heart cell, does not circulate to the heart, and therefore the special ingredient in wood causes no disruption. In fact other cells could metabolize that wood compound (break it into pieces) and render it harmless.
I'm going to riff on Tim B's answer and say it's about the vampire's regenerative process, which creates highly localized, intense heat. Normally this isn't a problem, but you add intense heat to wood, and you get... fire!
Once the fire starts, it produces the small energy kick needed by the initially-endothermic reaction to oxidize the rest of the vampire's cells. A chain reaction ensues, which is why the movies that get it right show vampires crumbling to ash when staked.
Even if you pull the stake back out very quickly (because whittling a stake per vampire takes too long) enough microfibers are still left behind to kick-start the reaction.
Edit: Of course this begs the flamethrower question (w/ a nod to Kaine's comment). If staking actually causes a vampire to die by fire, then why isn't a flamethrower a viable weapon?
Here it gets a little more complicated. First, when the flame is applied externally, there is the chance for some of the heat to dissipate into the air. Plus, those buggers are fast. So an initial blast of flame will certainly damage them, but they'll move out the the way fast enough to avoid the start of the chain reaction (and the movement itself will help dissipate more heat). Of course, if you could tie a vampire down and hit him with a concentrated blast of heat, then yes you could spark the chain reaction. But practically speaking you'd have do to that when he's asleep during the day, and getting a flamethrower past a vampire's minions is a lot harder than tucking a stake under your shirt.
The question sort-of implies that you're looking for the unique combat advantages of wood, but if there's no magic, it would have to somehow be super-poisonous to a viral infection, which would be pretty silly.
But! stakes were traditionally used to prevent vampires from reanimating; it would be driven into a corpse's heart as part of burial, or reburial.
The vampiric virus' goal is to knit the body together into a useful, mobile state. Stakes interfere with this:
Wood is cheap: burying a golden (or even iron) stake in every corpse is too expensive to be practical (in 1200AD). Metals would be stolen by grave-robbers; not wood. Wood is tough enough to last until the body is sufficiently decomposed to be useless to the virus. Wood is strong enough to pound into a leathery old corpse.
Only some woods that have antifungal and antimicrobial properties are necessary to combat the vampire virus.
This is the same reason that garlic repulses vampires, because of the antifungal/antimicrobial properties of garlic oil.
Also "holy water" is really just water that sits in a cold stone container for an extended period of time, which allows it grow a mold that has antimicrobial properties. It doesn't actually have to be blessed by a priest to kill vampires.
And the whole thing about sunlight is that the vampire virus changes cellular structure to create a massive increase of vitamin D receptors (VDRs) on the cells. When the VDRs are overloaded, the mitochondria actually produce an intense burst of heat, then the vampire burns due to the wick effect of its subcutaneous fat.
see wikipedia:
A little known fact is that you can also kill vampires with any oil rich in vitamin d, like fish oil!
Here is a specific type of wood that has anti fungal properties that actually does kill vampires:
https://link.springer.com/article/10.1007/s00226-011-0428-9
Also willow stakes are effective:
This was explained in the novel "I am Legend". Googling found that this detail was explained in chapter 17.
The reason stakes kill the vampires is that they allow air into the body. It also "keeps the flesh open so the body glue can't function". Normally any wound like a cut or bullet hole would be rapidly closed by that "body glue" explaining the vampires' durability. Penetrating the heart is unnecessary for this reason in that novel. Wooden stakes work better as they are more porous. Via this explanation, other things feasibly could work like an air compressor and pipe, but a metal spike would clearly work less well (as the glue would just seal around the wound).
As you can tell, vampires in "I am Legend" are a little different than other media and are somewhat reminiscent of zombies. Nevertheless, it gives a scientific explanation (sometimes good) for many of the vampire myths.
All of these other answers are trying to come up with ways in which wood could stop regenerative powers or why wood would be better than other materials. But that's not what the question as stated has asked. It asked if there is a scientific reason why a wooden stake is the ONLY option to kill a vampire. And for that question, you need not even consider how the wooden stake kills the vampire, you need only consider why literally every other means of destroying the vampire would not work.
And to that question, of "is there a scientific reason why a wooden stake would be the only way to kill a vampire?" the answer is a resounding "no".
There is no scientifically plausible way in which a vampire could regenerate from or remain unharmed from the myriad ways to destroy something. You can't have something that is immune to heat above a certain temperature, or immune to being cut up into little pieces and each piece being put into a box, or immune to pressure, or immune to every form of dissolving chemical reaction, or immune to sufficiently powerful explosives, or immune to starvation of useful energy, black holes, or, in the very long run, to entropy.
There might be some scientifically plausible way in which using a wooden stake to the heart is the easiest surefire way to kill a vampire, but that's not what's been asked in the title question.
When I first read Bram Stoker's Dracula, I was amazed at the highly rational, scientific way that the English heroes and their friend, Dr. Van Helsing, coped with the object of a Medieval Catholic myth in their travels. Van Helsing investigated Dracula in the same way that any Victorian scientist would examine a phenomenon of interest - say a diversity of finches on a distant archipelago or measuring the speed of light. Science is as science does.
So, scientifically speaking, vampires are Christophobic. They shun churches, holy water and crosses. Van Helsing "contaminated" Dracula's coffins by placing consecrated hosts in them.
In traditional Christian myth, all wood is considered sacred, because related to the wood of the Cross --- ergo, dangerous to vampires.
Is this a scientific explanation? If we are talking about vampires, this is as scientific as it is likely to get.
To me the big question is "why garlic"?
Wood is not necessary to kill, but it is the only safe substance to verify death.
You can kill a vampire with literally everything piercing the heart, but only wood causes involuntary spams which are quite dramatic (and cannot be faked) and only cease with death.
The problem is
so vampires could fake death if you don't hit the heart exactly. It only takes a few times for vampire hunters to know that approaching a vampire without knowing if it is dead is extremely foolish.
So if you hit a vampire with a wooden stake in the heart, (s)he spasms and when the spasms cease, you know (s)he is dead. The vampire also spasms if you hit him with wood into other body parts, but the spasms are weaker and can easily be explained by pain or nerve damage of normal people.
So many answers... A fun thought is that since the wood was once living, the virus "re-animates" the wood and grows a tree if it finds the right part of the vampire's body (the heart). Since the tree was so much more massive than the vampire, the energy required to re-animate the vampire gets sucked into growing the tree and the vampire remains motionless. It could make for a good sequel - the blood forest or whatever.
I agree with @Separatrix that the wooden part may be optional.
From a pseudo-scientific point of view, staking the heart may be used to stop the vampire from regenerating. As cutting the head off lets you eliminate the brain of the vampire and kills of the body, stopping the heart may stop the vampire itself.
Now, undead vampires are often described as having no pulse, but that makes sense only in a more "magically explained" context. From a more scientific point of view, all vampires still need blood to survive. It is only natural to suppose that this blood is carried around in veins and arteries.
Since blood is the source of power, stopping the heart would mean stopping blood flow - so movement, healing capabilities, strange vampiric powers ... the vampire itself may not die and fall into a dormant state, if you wish, but if it can't move it's almost as good as killing it (bonus points: it's easier to cut off its head). The important thing is that the vampire should not be able to remove the stake by himself.
Bonus points, maybe wood also poisons the blood, making the heart stop faster or impairing movement better on elder/more dangerous vampires.
I am going to take some liberties here with the concept and definition of 'vampire'. Apparently, from what I read, it has been clarified that you are NOT speaking about a 'generic' vampire, but a specific concept that can be modified to suit the requirements. That is, a human body that has been taken over by a virus, which is somehow able to maintain homeostasis and muscular function in the host body without the host body itself being alive (else killing the host body would kill the 'vampire'). I am also assuming that such concepts of 'sentience' and 'memory' in the host are moot points. More a zombie than a vampire - or a cross between them. It is having a moving, semi-functioning body that is important, not any other human characteristics. It, of course, begs the question be asked 'If you cut off the head of a vampire, does it still retain its memory and personality?' I am assuming the answer to this question is moot. The virus somehow becomes the personality, and if there is a brain left in the host, than the virus can somehow manage to keep it functioning in some non-living way enough for the virus to 'read' it. But, of course the destruction of the host memory and personality would not result in the death of the virus.
In which case, the question is not how to kill the vampire host, but how to kill the vampire virus. Since a virus can spread throughout the host, it would imply that any part of the host that remains could potentially contain live virus, which would still be able to reproduce and 'control' the host. That is, it is the virus, and not the host, that actually lives on. This would assume that if any small part of the virus lives, it can propagate again to 'infect' the entire body, or whatever is left of it. A walking hand, where the muscles are systematically triggered by some biological process of the virus? - the virus can either trigger neural impulses in nerves, or it can send out chemical messengers.
So if sunlight and garlic were toxic to the virus, they would result in the death of the virus.
The question then becomes 'How else do you kill this virus?' Certainly, any sterilization process that can be guaranteed to eliminate 100% of the virus, but we know there are viruses that can survive almost every physical environment, albeit in a dormant state. There are viruses that can live dormant in the human body until the body dies, and perhaps even after. Just one virus in one cell of the former host could potentially re-infect.
So why just a stake in the heart? A particular reaction between some protein or other in wood and some cell structure specific to the heart, that produces a toxin that will kill the virus. Perhaps it could even be another virus that lives only in the human heart, that would survive the death of the host, just like the vampire virus. This would assume the circulatory system still functions to some extent to spread the toxin.
This could, perhaps, be molded or refined into some concept of a virus that only lives in the human heart, being the only thing that could produce a toxin under specific circumstances that would kill the 'vampire' virus. It is a solution that does not require the host to be alive in any sense of 'human sentience', or of the heart working as a heart. A dead heart could also host the competing virus.
It does, however, seem to imply some form of sentience for the vampire virus itself.
You can't get a plausible scientific reason other than "scientific" one.
There are many ways to kill. If you put vampire in a high temperature environment, it will burn(assuming carbon based living organism).
If your vampire isn't made of carbon, you can still melt him, crash him, and depending on his body - suffocate, dismember, poison, dissolve with acid, deadly injure, or place him next to strong magnet which would disrupt his cells.
There is absolutely nothing unique about wood that wouldn't make it hilarious when confronted with above methods.
So, either don't try to explain why and just make it a well-known fact that does not need to be explained, don't use it as a (sole) way of killing the vampire, or come up with any explanation and call that a "science" in the real world(which you may read as "use handwavium"). Some of the reasons might be:
Although there's really no way to do that without using pseudoscience.
The science behind vampires is that they are actually people who have gotten rabid. Sensitivity to light, aggressive behavior, Intensified desire for lustful passions, fear of water (note water does not actually kill them - hollywood myth), don't eat solid food, (difficulty in swallowing solid food), but do bite others of their own species, don't sleep at night.
While the stake through the heart may kill a rabid animal - It does not kill the infection. If the blood of the vampire gets consumed by another animal that next animal will get infected, (about 21 days after exposure) ... thus becoming a vampire. Although most the stories are that the individual vampire himself, at least so far, does not come back to life.
According to Hydropathic Encyclopedia, Robert Trall, M.D, Garlic can stop rabies / the vampire infection. Garlic or muriatic acid should be placed on the wound.
Since 21 days after, (the stake was suposed to have killed the vampire), a new victim appears clearly the vampire escaped death. How else in the 12th century could he have another victim if he was dead.
The vampire must be burned to kill the infection. If we are to consider the science behind the myth. And, yes rabies was present in the populations when the myth started.
Yes, bats can carry the vampire infection, but scientifically vampires can be almost any mammal. A vampire that was only killed though the heart can return as a wolf, if he was not burned.
Possibly the common vampire has a rapid healing factor and regeneration which does not work for the heart. (Either by not being able to heal damage to the heart or by any damage to the heart causing massive blood clots and/or tumour like scar tissue growth.)
A wooden stake could deliver splinters into the heart causing infection and exacerbating the problems.
Personally, I prefer the vampire of Arcane origins BUT, I'll have a go at this.
The only thing I can think of that could scientifically justify (as @Rekesoft so perfectly put it) "someone who has become immortal, highly flammable under sunlight, allergic to garlic and cross-phobic due to a viral infection can only be killed through a stake in the heart, but only if it's made of wood" is by a virus of intelligent design!
It's maybe something you'd struggle to justify any of your characters even HAVING an explanation for, but if a sufficiently advanced - and probably weirdly religiously inclined - race, be they alien or ancient earthen, wanted (for whatever batty reason) to give mankind (or some portion of it) some weird superhuman abilities and very specific vulnerabilities, a very sophisticated DNA altering virus may be the best/only option. The specific method(s) of killing a vampire (e.g. a steak to the heart) could even be deliberate, hidden kill-switches (just in case). In which case, the stake causes the largely dormant virus cells to trigger a violent pyro-self-destruct mode.
This could even explain some of the less well known 'myths' like how stakes made from ash wood are more effective 'just because the virus recognises them quicker', or that vampires are obsessively compulsive about counting and untying knots 'because crazy ancient/alien cultists tried to make mankind act more appropriately' in their own mixed up way. Or you could brush those things off as unintended side effects.
Maybe the creators of the virus got bored and left because 'our work here is done' or 'whooee, well that didn't go well, lets try somewhere else', or maybe they were destroyed by there own creation due to an unexpected shortage of timber and heavy overcast from volcanic clouds.
The following could be used as a plausible scientific explanation.
The Vampiric virus causes the body to produce a new enzyme that gives the victim all the traditional vampire powers. This enzyme, though, causes the skin cells to have a violent flammable reaction to sunlight. The enzyme also combines badly with allicin from garlic or cellulose from wood to form a compound that is deadly to the body. As allicin is partially aromatic, it's presence in the air can be deadly to the vampire when breathed in. Cellulose in wood, however, is only deadly to the vampire when inserted into the body. As the heart is the primary organ of storage for the vampiric enzyme, this is where the wooden stake would be most effective.
Another option is that wood is somewhat porous, while metal isn't (as much). The regeneration can work around and eventually push out a metal object, but enough body material gets absorbed by the wood that the regeneration process embeds the stake further instead of pushing it out. This is only an issue at the heart where there is enough blood to sufficiently soak the wood; elsewhere the body is able to eject the wood before it's gets embedded by regeneration.
Vampires are very clever and have the same drive to survive as anyone else, and a wooden stake to the heart is actually one of their reproduction options
One possible reason for wood being the ONLY type of stake material to work is simple propaganda put out long ago by the very clever vampires themselves.
The vampires became aware of their own vegetative propagation ability by discovering quite by accident that they have a type of meristematic cell ( https://en.m.wikipedia.org/wiki/Meristem ) biology more closely related to plants than animals. This was discovered through noticing repeated occurrences of a wooden staked vampire eventually regenerating many weeks and sometimes months later, whereas none of the metal, bone, or icicle staked vampires ever regenerate. Through experimentation, they found out their hearts just so happen to have the highest concentration of their meristematic cells, so naturally that's the best place to 'plant' the regenerative wooden stake. Like other plants, they also have other less taxing methods of reproduction, but this one works in a pinch.
It didn't take long for them to launch a sneaky misinformation campaign out to the non-vampire community (see other excellent answers for good propaganda examples) to ensure active vampire hunters use wooden stakes that will likely allow their eventual return from the grave.
As anyone who has worked with plant propagation knows, this isn't always a fool proof method of reproduction, but considering it is actually taking an enemy's intention to kill them and turning it into reproductive assistance, it's really rather ingenious.
Take your pick - ;-P
Vampires Are Not Real
If a vampire can only be killed by wood, the obvious question is why doesn't being put through a woodchipper and then baked at a thousand degrees kill them. The obvious answer is that not life form we have found could survive that. Then we may conclude that vampires don't really exist. Never-the-less, vampires might seem very real to the protagonists, and the idea that they don't exist may seem absurd.
"But I've seen Vampires!" Some parasites change the way the host behave. Perhaps a parasite induces hallucinations that manifest based on the desire to see one dearly departed, but corrupted by creeping dread.
"But there have been many witnesses of the ghouls the vampires have created, and their stories agree, so they can't be a hallucination!". The parasite sometimes also induces ghoulish-behaviour.
"But the ghouls had bite marks!" The pair of marks are the entry and exit wounds from the parasite.
"But an entire town was butchered by Vampires!" Sometimes the Parasite induces violent behaviour, sometime a Vampire cultist hopes that killing and drinking the blood of the infected will give them immortality, and sometimes some serial killer just decides play the part of a vampire for some reason we will never understand.
And most importantly
If vampires are hallucinations that means that although the universe is science based they may appear like supernatural spiritual things that are too horrifying to obey our physical laws (at least until their real cause is discovered).
