The Elves were not granted the Gift of Death (as they called it), so yes, they did not die of old age. However, beyond sharp eyesight,
lightfootedness, and some astounding reflexes, they were just as vulnerable to mortal wounds as humans, and despair could drive them to death.
But even then, the death they did experience was little akin to humanity's. Look at Glorfindel, for example. While some could argue that there were actually two Glorfindels (one who fought Gothmog, the other present during the War of the Ring), the fact that no other Elf in the entire mythology has the same name as any other Elf makes me believe that they're one in the same, and that he was just the first (or one of the very few) Elves to choose to come back to Arda.