Why do elves eat?

[Tuor Turambar,Cursed by the Valar]

I'm sorry, but it is not true that elves can only die of three things.

orrect me if I'm wrong, but isn't it true that elves only can die from three things (battle,grief or getting tired of the world and leaving their bodily attachment to the it)

Elves can die of anything a human can die of, except old age and disease. They can also die of growing weary of the world, which humans cannot do.
Elves can die from fire, sword, drowning, starvation, suffocation, poison, ect.
There are many interesting threads on the topic of how elves can die. You might want to search and visit them.
Hope this helps.