Why should elves be immortals ?
They are long lived. So long lived that for a human, they seem immortals. In France we have a saying that goes:
"De mémoire de rose, on n'a jamais vu mourir de jardinier"
(In a rose's memory, gardeners have never died).
That's enough.
Besides, I'm not fond of "Tolkien's elves" in a D&D context, because "Tolkien's elves" always refer to "elven heroes like Feanor, Galadriel, or at least Legolas" rather than "base elves like those that get drunk in the Hobbit".
I've made this rant countless times on the Wizards board. All the überness of elves that people want to be enforced on all elves "because they're elves" in Tolkien, comes in fact from "because they have the seen-the-Light-of-Valinor template". A simple dog who had seen the Light had killed Sauron's werewolf captain. Yet noone whine that all dogs should be able to kill werewolves just because they're dogs and they do that in Tolkien's books.