Humans are savage creatures. This is indicative in the way they get rid of bodily waste. Squatting over a glorified hole in the ground or aiming their spear tip toward the same hole and hoping for success is a matter best left to barbarians with no dignity or self respect. For this, you are to be pitied and belittled. We elves naturally developed a method that is more worthy of our nature. Our bodies are far better at breaking down our food, leading to cleaner digestive systems. What little waste remains however, can't just sit there, otherwise it would poison us. How can it be removed without exiting the anus?
Pooping is far less disgusting than any other solution if you think about it my dear arrogant Elf.
The most natural method as an alternative to Pooping would be puking like some other lower life forms. I havent looked into Elf Defecation but considering your self-important race thinks Pooping is disgusting I expect not.
I see one of my esteemed colleagues has suggested the possibility of locking your waste products in hair, but here the gross part starts. Humans eat and do not absorb materials from their food that is waste anyway, then add bodily wastes to that which they poop out. If an Elf grows it in their hair, that waste has to go into their bloodstream and slosh around their bodies until it finally is collected and build in their hair. It's a dirty process that would cause many bloodborne diseases and vasculairy problems.
And what about liquid waste? Humans pee it, and a small portion is expelled in sweat. If your race does not pee, all it has left is to sweat it out all across their bodies, or pant like a dog if not to spit it out, only to have traces stick to their skin and mouth.
No I do not think even Elves are as filthy to forgo the simplicity and cleanliness of standard defecation, otherwise I would refrain from ever coming close to any Elven seers I meet, as their repugnant smell and disease ridden body would warn me long before I would have the guts (hah) to meet one face to face. Due to the lack of stories about constant disease and stinking filth surrounding you Elves, a racial taboo on the entire process is all that I can name to be the culprit of such nonsense that you do not use an anus to defecate or bladder to rid yourself of pee at an appropriate moment.
Wastes are incorporated into hair.
Wonder how the elves accomplish their flowing locks? How an elf seems able to try a new hairstyle every week? It is all about the hair. The elven keratinocytes lock up indigestible fiber, waxes, and other materials delivered from the digestive tract and sequester it within long flowing tresses. This added material means that an elf will need to cut his or her hair every few days or adopt long Rapunzelesque braids.
As for nitrogenous wastes, these too are expelled by glands associated with the hair, giving it that glowing sheen. Commensal microbes on elven heads partly metabolize these wastes, giving elves their characteristic smell of caramel and violets.
If you insist on no solid waste you would need to break undigestable molecules down to liquid or gaseous forms.
A major problem if you want to avoid any excrements is that you would have to break fibers and other undigestable contents down completely until the molecules are small enough to be assimilated by the body and then expelled through the breath, sweat glands or urine. These fibers leave the human body undigested because breaking them down is either impossible in our gut (due to missing enzymes) or needs more energy than can be extracted from them.
So what do elves do instead?
Their digestive tract doesn't end in the anus, but curves back and attaches to the lower end of the esophagus. There it forms a little pouch-like organ, just like the stomach, but this one withdraws even the last traces of water from its content. Due to the highly efficient digestive system, only a tiny amount of waste ever gets into that pouch, building a thin layer over the waste already contained in it. They practically produce pearls of their bodies waste.
When the pearl gets big enough, it's transported up the esophagus and spit out. In true elven fashion they either sacrifice it to the earth as fertilizer or literally sell sh*t to those stupid humans.
We elves naturally developed a method that is more worthy of our nature.
I would modify your premise, which I will get to in a moment.
First, we humans preprocess our food to make it more edible. We peel it, we core it, we throw away the seeds and stems and leaves, and often skin, bones, organs and any other parts we don't like. We cook it, to make it tender.
Elves that thought elimination was not worthy of their nature have simply taken this a step further: They prepare their food to eliminate waste before they eat it. They have a superior digestive system, but still, they remove all indigestible components from the food they eat, and if that is not possible, they don't eat it! The same thing for drinks. The method they developed lets any excess liquid they ingest be quickly evaporated from their sweat. They limit their intake of fluids to match their capacity to sweat.
It would be like getting all of your calories from honey. Their pre-processing and cooking method produces food with no residue.
Also, much waste in the human body, like toxic chemicals, goes to "garbage dumps" other than the colon or bladder: Ear wax, sweat, tears, hair (on the head and on the body), finger and toe nails, skin. Any microscopic indigestible waste could do the same; it emerges as a thin film on their body with their sweat that is washed away whenever they bathe.
Elves do have anuses, it's just that they only have one common use with human ones.
For excretion, though - elves have developed a symbiotic relationship with an insect, a dog-sized variety of phasmatodea. The insect is carefully (for dwarven standards) shoved up the elven cloaca, where it can eat and drink whatever is there. This has the benefit of not requiring a sitting position from the elf, so while they mostly prefer to do this while lying down on a bed, they are free to walk around with a monster hanging from their derrieres if they want.
You know how every joke can have its truths? When pilgrims and explorers say that an elves have a giant stick up their [be nice], they are being literal. They probably witnessed, during their travels, an elf walking around during their daily cleanup.
P.s.: this has been featured in (serious) sci-fi literature before. In Arthur C. Clarke's Rama series, there are aliens who do shove bugs up each other's [be nice] so that the bugs can eat their excrements while inside.
You just have a rather more advanced version of the owl's digestive system. In owls, hard & indigestible parts of the prey are formed into pellets, and vomited up as "owl pellets": https://www.carolina.com/teacher-resources/Interactive/basic-information-on-owl-pellets/tr11103.tr Your elves expand this to include all the waste.
Though I'm not really sure this would be considered more dignified than the way humans do it :-)
Dryadism (is that even a word?)
The Elves have evolved a symbiotic relationship with their grove. Every now and then, they will either lie in the glade, or be swallowed up by their home tree.
In both cases, tendrils from the grass or tree will enter the pores, and locate the apovlitocysts - the deposits of waste material accumulating beneath the dermis - through chemotaxis. There, they'll inject a cocktail of enzymes that liquefies the deposit, allowing them to suck it dry. The plants will also supply vitamins and feel-happy alkaloids in exchange for the bounty of readily-assimilable nitrogen and nutrients they're offered, not unlike the Nepenthes lowii that supplies its customers with sugary nectar. The alkaloids have the primary purpose of getting the elves relaxed and in a stuporous state.
Not that the Elves are aware of this happening, since the gory details are even more disgusting than the human solution. They only know that with the passing of time they get slower, irritable and more tired, like a human with sleep deprivation; and the communion with the grove leaves them rested, refreshed and rejuvenated. And, of course, staying too far away from the grove for too long kills them.
The relationship evolved from a predatory one in the depths of time - the plants had already evolved the "get them high and kill them" trick against small animals. The effect on the elves' primate ancestors was much less due to their greater mass, and they mainly got a high from lying in the carnivorous grass. So they started doing so. The plants had to be content with absorbing superficial matter, dead skin, oils and the like; but they evolved tendrils to scavenge more and more, while the elves evolved to avoid the ill effects of the tendrils themselves. Down that road, the necessity of expelling the matter accumulating in the pores - a necessity which elves shared with humans (there are several disgusting videos on YouTube promoting 'beauty treatments') - grew less and less.
Eventually, all mechanisms to drain pores to the outside atrophied, outsourced to the grass. Proto-elven organisms then started finding more and more convenient to use subdermal cysts to get rid of waste material, where it magically disappeared after a while. A couple million years later, the elves' lower intestine is little more than vestigial. At the same time, their diet is somewhat restricted in quantity and nature (they make up for it with lengthy and elaborate preparations that put to shame humans' nouvelle cuisine); were an Elf to eat large quantities of indigestible matter, he would get spectacularly sick.
While rich in sugars, base proteins and water, elven food is not automatically digestible by humans - to increase their digestive efficiency, elves also evolved a commensal symbiotic gastric flora that can break down most vegetable matter and complex carbohydrates. On the negative side, erumëloirë is an Elven poison that acts by killing off most gut bacteria, leading to a condition similar to wasting sickness from which recovery is long and difficult; this class of poisons have almost no negative effect on humans, who call them penicillins.
I'll try to do this delicately. They have a plant beside their bed that extracts the excrement nightly for fertilizer. They never have to deal with it themselves. The plant applies a neurotoxin that deadens the entry and removal sensations. They wake up in the morning refreshed and empty. When traveling, they carry a seedling that performs the same function. Over the course of history, most if not all have forgotten how this works. They simply keep the plant nearby due to tradition, or as a good luck charm, or as a constant connection to nature. The symbiotic relationship is forgotten, but haughty attitude and physiological features remain.
It is hard to reconcile “the wee little people” of legend and the sharp-shooter tall-blond-and-handsome elves that we meet regularly. The answer is simple: elves are born quite tiny (within acorns, frequently). They excrete nothing. The hard matter that cannot be digested is embedded in their bones at astounding densities. If you’re going to have dexterity of an elf, you need bones that won’t shatter when you jump down two stories or leap tree-to-tree. It takes a century of eating to get enough material for human-size bone structures, but elves are long-lived creatures.
The elves are full of it.
It's true that the elven anus is pristine. In fact, it's part of a secondary respiratory system that's used to bring oxygen to the digestive tract to allow them to process food using aerobic as well as anaerobic bacteria, which is part of the reason they produce much less waste in the first place.
Being elves, they also use the anus to produce music of unearthly beauty, of course.
But, contrary to what they tell the other races, elves do produce waste. Not nearly as much as humans or dwarves (plus, they don't eat nearly as much in the first place), but over the centuries, it builds up. Because they have no way to dispose of it.1
Elves are immune to the ravages of age, resistant to all diseases, and difficult to kill in combat. As Tolkien explained, elves usually only die by fading away willingly, when, after long millennia, they've "lost the will to live."
Which sounds nice when an elf says it, but what they really mean is that they're sick of living with constant severe indigestion.2
Why do you think those great elf lords always have a pinched face and a curt manner whenever you speak to them? They'd have you believe it's because they find it distasteful to talk to someone as lowly as you, but the truth is, they're the same way with their spouses, children, and co-councilors. They're perpetually in a bad mood because they're perpetually uncomfortable, and it gets worse with every passing century.
1. Elven legend—rarely shared with the other races—has it that after death, their spirit can be reformed into a new, waste-free body. And, in fact, there are a few apparent cases of ancient elves returning from the dead. With the wisdom and experience of millennia, but the digestive tract of a newborn, they tend to be unsurpassed both as warriors and as dinner guests. See Tolkien's description of Glorfindel for an example.
2. Tolkien even describes it as "wasting away".
As mentioned in the question, the Elven biology is exceedingly efficient at processing food and drink. However some substances are so toxic, that they can't be completely processed and recycled. These accumulate (at a very slow rate) in small glands in the forearms, and thence to the the elf's fingernails. It takes on average three years for these sacs to become full to the point where the waste must be excreted, however they can be released at will when needed (or occasionally, autonomously in high adrenaline situations) when an elf is forced to fight while unarmed. While an elf-scratch probably won't be immediately fatal, it will be exceedingly painful, is much more likely to become infected, and almost immediately causes dizziness and fatigue.
Some elven cultures make it a point of pride to keep their toxin sacs empty, releasing and cleaning at least once a month. Being from a "higher society" they have no need of the primitive defenses of their forebears. In some other, more violent cultures, they do their best to keep their sacs full all the time, so that they can scratch at a moments notice. These elves wear gloves when they eat to avoid poisoning themselves from their constantly tainted nails. Most elves however fall somewhere in between, getting a yearly manicure were the toxins are drained.
I was very tempted to just give the joke answer of "just have them vomit it up, duh!" but I'm not going to do that. Instead, make it so their digestive systems break everything down to a fine liquid (if you want more detail on how to do this, make their digestive systems have the right enzymes to reduce anything that goes in there into a liquid. This liquid is then spread out extremely thinly and leaves the body in their sweat.
Elves can simply just have a more efficient digestive system. Encyclopaedia Britannica gives a breakdown of what feces is composed of here.
Normally, feces are made up of 75 percent water and 25 percent solid matter. About 30 percent of the solid matter consists of dead bacteria; about 30 percent consists of indigestible food matter such as cellulose; 10 to 20 percent is cholesterol and other fats; 10 to 20 percent is inorganic substances such as calcium phosphate and iron phosphate; and 2 to 3 percent is protein. Cell debris shed from the mucous membrane of the intestinal tract also passes in the waste material, as do bile pigments (bilirubin) and dead leukocytes (white blood cells).
Humans, can't break down the undigistable food, but that doesn't mean other organisms can't. Basically, elves would just need a source of enzymes like cellulase and pectinase. Whether it's symbiotic with other organisms like bacteria, or a gland that produces if for them is up to you. Likewise, the other stuff like fats, proteins, and inorganic substances can all be taken care of with a more efficient digestive system. At that point, all you have to deal with is the dead stuff your body produces (bacteria, cells from the intestines, etc). This can be handled a couple of ways: maybe elves don't produce as much of this stuff. Or maybe instead of having the digestive track end with the anus, have it end with an organ (essentially a second stomach) specialized in breaking these things down.
Once the solid matter is dealt with, the water is not necessary for a bowel movement, so their body can use it for other things. That means elves need to consume less water to survive (major advantage), and if they do consume excess water, they can just lose it through respiration or perspiration. Any excess chemical elements that their body doesn't need could also be gotten rid of this way.
Also, if they can break down things like cellulose, then that means they need to eat less plant matter to survive, and opens up new potential food sources for them, making them an even more efficient organism than humans in yet another way.
Go with a subversion. Elves are being arrogant and will claim that the concept of the "bath room" is utterly disgusting... but will not detail how elves would take care of such business as it is undignified to speak of.
When we finally get lost for a few days with an elf, we find that they do a number two, too. It's just such a taboo subject to talk about they hide their water closets from even their significant others... oh, and not only does their... um... stuff... not only does stink, but it's worse than humans. It reeks! Courtesy of a high fiber diet, your bathroom is now inhospitable for days... and the elf forgot to turn on the ventilation fan because you didn't get that far in your explanation. And he most certainly didn't wash his hands!
Why is he claiming his way is so much more refined and better than ours? Well Elven rest rooms all come fully stocked with the very best in hair care products... which is the whole reason they don't talk about that... it's rude to ask you how much money is in your bank account too. That's the real reason they are so dedicated to you not finding out! They didn't want you to know where they kept all the lotions for their luxurious locks, because as the TV told them, they are worth it! They didn't even realize the stench was putrid to us mere mortals.
It's a well-kept secret that elves are able to crystallize their waste similar to how an oyster creates a pearl. When an elf feels sufficiently backed-up he or she will have an instinctive understanding of what type of mineral should be seeded via their posterior end to start growing a crystal that will accumulate the majority of the waste in their system. Once the crystals have grown to sufficient size they are removed via the same orifice and put to use. Ever wonder where the elves get all those pretty little crystals they light up their tree houses with? Or where all those powdered "mana" crystals they mixed into that mana potion you just chugged come from? The truth is a good portion of elf culture, and their exports, revolve around the fun things they do with their crap. The next time you see an elf with a particularly pinched face just keep in mind... chances are it's probably not you, they are most likely just in the process of growing a "big one".
Elves don't excrete because their appendices are actually pocket dimensions from another plane of existence. The waste simply builds up slowly over time, and eventually, after thousands of years, it becomes full, at which point the elf shows signs of declining health and ultimately dies of terminal constipation, and the appendix disconnects from our plane, where ultimately it will generate a demon-like being.
A little-known fact is that the famous magical items like the portable hole and bag of holding are actually crafted by extracting and preserving the appendix from the corpse of a young elf.
Elves still poop... but their poop is not gross.
What really makes poop gross? It is not where it comes out of our bodies but the fact that it smells bad, carries diseases, and has a bad habit of squishing unpleasantly underfoot. If an elf intestine where to fully dry out poop before ejecting it, then this would kill off all the odor causing bacteria, resulting in something as unoffensive to others as a rock. Ofcourse, you don't want to have to force a hard, jagged poop out of your anus, so you want to make sure it's as smooth and polished as possible. So sure, an Elf still has to pop a squat just like the rest of us, but while our waste is something no one want anything to do with, Elf Poop is so nice, human jewelers go out looking for the stuff.
Symbiotic Wood Armor
Elves are born with larger pores over their vital areas. Elves are born under the mother tree which is sacred to their species. The tree has poppy-like flowers that produce small seeds and when a baby is born, there's a ritual to cover the baby in these seeds. The seeds are small enough to fill in the pores, it is seen as a blessing -- an offer of protection from the mother tree.
The elf's pores excrete sweat filled with nutritious waste that the plant can use to slowly grow. It forms tiny roots out from it to hold itself in place, but then grows outward and forms a hard shell to protect it's source of sustenance. It grows as the elf does, hitting maturity at the same time the wooden armor becomes dense enough to be a solid protection.
It's at this time they are able to leave their sanctuary and venture out into the world. A common practice would be to stain or lightly carve the wooden armor. There are master carvers that devote their lives to this. Your family's crest is carved into it when you reach maturity and your loved one carves their own design into it upon marriage.
Solid waste consists mostly of material that cannot be digested and was never absorbed into the body. In order to not expel undigested matter, these elves must be careful to never ingest indigestible matter. They only eat carefully measured amounts of highly processed (perhaps magically processed), liquid food. Many elves subsist only on the nectar of flours they cultivate.
All material elves eat is absorbed into the blood. (because of this, Elves get drunk very efficiently). Waste is excelled as a gas by breathing, and as a liquid by sweating. This is what gives elves their characteristic sweat and sour smell that humans find so appealing.
Because of this, elves cannot tolerate human food. If forced to eat what humans can prepare they usually opt for water with honey. Even though that does not account for all of their needs and after several days of the substandard diet they get quite ill.
What little waste remains after digestion is excreted through superficial tissues, like hair, toenails, and the skin.
Elves are so much better at breaking down food that they can produce new cells at a faster rate. Instead of accumulating in the guts, the residual waste is broken down in small pieces and distributed all over the body. where it can be used as cheratin for hair and toes, or accumulated in the dermal cells.
Since dermal cells are built up faster, they tend to be shed faster also. This is not gross as a snake skin-change, as it doesn't happen at the same rate in every point of the body, but still Elves are more prone to the occasional scratching - their fallen cells tend to itch more easily, and maybe, despite their long, beautiful hair, they are more subject to dandruff also.
Have you noticed how Elves seem to be obsessed with hygiene? I mean, I bathe myself once every two weeks, and they still look down at me! Anyway, I suspect that their regular baths (one a day to keep the darkness at bay) helps them get rid of the excess skin, that is, after all, waste material.
In humans and other being evolved from animals with a digestive tract, there is an entry point for ingestion, and one (or more) for excretion. And extraction+transformation of input matter is done over the length of the tract.
Elves are not evolved from humans/animals. They are spiritual beings who look, function similar to humans from an external perspective but do not have the same inner functionality since they descended from ether/spirit.
Therefore, the matter that they consume is turned to spirit. i.e, matter-energy => spirit-energy. However, different rules apply in this conversion as compared to matter-energy => chemical-energy. Just like digestion initially requires energy to break down the matter to chemicals (endothermic process), the conversion of matter to spirit also requires the elf to expend some amount of energy initially.
Matter that is closest to nature (plants etc.) and made of the simplest molecules are therefore contain the most spirit-energy per unit matter. Therefore Elves would favour plant based foods (vegetarianism).
The elves generally live near places with a lot of water. This isn't a coincidence. This is because they excrete all waste via nr1.
In humans, if the digestive system is upset, for example by disease, then it can happen that one of the main functions of the intestines is disrupted. This is the uptake of precious water. The result? Diarrhea. If the disruption is large enough it can be practically fluid with only tiny pieces of excrement.
The next step is to combine this with some anatomy of birds. Birds 'toilet visits' contain both kinds of waste from one orfice. The difference here is that the elves have specially adapted bodies to release nr1 and nr2 at the same time. Because of the watering down they go more frequently. As a result you need to prevent dehydration by drinking a lot more.
Maybe that us why elves are often depicted as tall. This is mostly storage space for all their waste, so they don't need to go too often to the toilet.
Do note that doing both from one orfice does require a more elaborate immune system at the specialised excretion organs. They aren't mixed in many animals for a reason. For males it also serves an extra purpose, so the plumbing gets very complicated.
Elves don't undergo something so messy as childbirth. Instead, they lay eggs beneath the roots of great trees. Every once in a very great while, one hatches, and the new Elf is cherished.
But all the others are simply waste wrapped up by the shell gland. It's neat, and it may deter predators that would otherwise feast on their next generation.