This is an incredibly broad question. Listing the reasons two peoples can hate each other could fill volumes. In our history of humanity, we have found quite a few reasons to hate each other.
Now Stack Exchange frowns upon questions that result in "list answers." At first I thought this question had to be too broad, and worth closing, because it had to generate a list. But then I got to thinking, the question of "why might two people hate each other in general" is actually quite the useful World Building question. Accordingly, my answer is very generic.
For discussion purposes, let's say every individual, or group of individuals, has a "goal function," that they use to decide whether a potential action accomplishes their goals or not. This goal function can change over time, as people's opinions change. With this concept alone, we have a powerful tool for capturing enmity between two races, for we have a feedback loop. Our goal function affects which actions we take, and our actions lead us down paths which shape our goal function. We've all heard the effect of feedback. The effect of a slight whisper into a microphone that's turned up too loud feeds back from the loudspeaker to the microphone until it rapidly turns into an earsplitting sound.
Usually these goal functions directly benefit the individual or group. "Planting seeds ensures I get to eat when winter comes, thus planting seeds is good," for instance. Wanting to build up your own quality of life is generally considered to be a healthy goal function.
However, when dealing with other minds, it's not always easy to figure out what those goal functions should look like. If Dragons' planting of seeds, food high in protien to replace the carnivorous faire of their ancestors, involves slashing and burning sacred Elf forest land, we can get contorted goals to deal with it. The Elves may decide, "Dragons cut down our forests. They don't respond to reason, but do seem to respond to pain, so if we hurt them when they are cutting the forests down, maybe we can teach them how to not cut down our forests, despite the language/cultural barrier." The key to this is that the elves have made their "goal function" include a conditional "opposing Dragons achieving their goal function when they are cutting down the forests."
At this point, the Dragons can short circuit the feedback loop. If, at this time, they decide that the Elves are probably trying to tell them something about their farming methods in their own special way, and decide maybe the best way to solve this is to keep the Elves interests in mind, the feedback loop is broken. The Dragons have decided to make "supporting the Elves achieving their goal function, because it's good for the Dragons."
However, more likely, given the lack of cultural and linguistic interaction that caused the problem in the first place, the dragons will respond as a child does. Rather than getting the message of "don't burn down the forests," the message they receive is "don't burn down the forests when the Elves are watching." They get creative, finding ways to sneak things in. It's only natural. They may have an exploding population that needs food, and thus farmland. Maybe they don't want to turn to eating the Elves, as their ancestors once did.
The result is quite agnostic to the rationale of the Dragons. The Elves only see the effect. It's getting harder and harder to actually catch the Dragons in the act. The forests are still getting destroyed. At this point the Elves engage in what I believe to be a key step in the cycle, highly visible in the aftermath. They make the decision that, since they know Dragons are harming the forest, they declare that they will punish the Dragons, even when they don't catch them in the act. Any Dragon seen will be assumed complicit to the burning of the forests. The goal function shifts from "opposing Dragons achieving their goal function when they are cutting down the forests." to "Since we know the dragons are cutting down the forest, opposing Dragons in any activity is good."
The key word in that goal function is "know." When we use that word in a context such as this, it typically means we're no longer looking to change our mind. Perhaps more to the point, we no longer assign value to putting ourselves in good positions to test our "knowledge." Maybe it was all a misunderstanding. Perhaps the Elves know this, and recognize that it may be a misunderstanding, but they're simply no longer going out of their way to find more information to bring into the equation. This is essential for animosity: the loss of a desire to act towards finding a common ground. You may be happy if a common ground finds you, but you're no longer inconveniencing yourself to find it.
From the Dragon's perspective, this increase in hostility is frightening indeed. Dragons are getting wounded and even killed by Elvish flash mobs. They have to act. Going back to our child reasoning, the elves have shifted from acting as a mother spanking the child with a wooden spoon during the act to a father spanking the child with a belt after the act. Now there's debate about whether mother's wooden spoon helps or hurts a child's maturation, but its generally agreed upon that the belt does not. If the punishment comes in slow enough to permit intellectual analysis, it will be rationalized upon. The Dragons respond by changing their fitness function. Clearly supporting the Elves' desires is detrimental to their own. They sever the last thing holding the feedback loop at bay: their goodwill towards the Elves.
Now, many things can happen. The feedback loop is unimpeded -- the sound tech at the board is away on break, and the microphone is primed to screech as loud as it pleases. War is a possibility. The key to war is the point where the harming of Dragons becomes equal or greater in value to the lives of the Elves themselves. If that hatred can rise to that point, war will occur.
However, it does not always reach that point. Distance can help. Not just physical distance, but mental distance as well. Perhaps the Dragons start building a wall to protect their farmland. Now there's a clear mental barrier between "us" and "them." With this barrier in place, it may be harder to convince Elves that its worth dying to kill Dragons. In this case, the feelings don't go away, they simmer. These boil down into the enmity between races.
Enimity is not free. For most species, children are not born hating anything. They must be taught to hate, and they will raise questions along the way. Something has to provide the energy required to maintain this hatred as a few young individuals question whether maybe there's a more peaceful way. However, the Elves may find that a little nugget of that boiled down hatred is easy to transmit to the next generation. Even though most of the anger and rage may die down over time, this little bit procreates from generation to generation, ensuring an apparently built-in natural hatred of the Dragons. Long after the anger dies, long after the wall is torn down because it's no longer suiting anyone's needs, the hatred may propagate.
And its very difficult to face hatred with love. It takes skill and inner strength. Depending on how strong the Dragons are, they may find this enmity from the Elves bothersome, and match it with an enmity of their own, home grown from Elvish insults.
And now you have a "natural enemy." It may not even be evident that it's part of your society. It can also simmer this way for an extraordinarily long time. Often this hatred permits certain actions which benefit your race, such as an increased tendency to take from others. If the hatred becomes the foundation of the good, it can be very hard to remove this natural animosity.
I have shown a rather large example of this, but history has shown that such feedback loops can occur as a result of mere personal insults to leadership as well. Two tribes may stay enemies for a long time after a spat between tribal leaders.
This also suggests the resolution to such enmity. One needs a feedback loop inspiring people to come together, instead of driving them apart. If it is strong enough, it will begin rooting out the hatred. But that's for another world. You want hatred between your two races, you can get it!
