According to the LotR wiki, there were at least 4 sets of Elf twins.
- Amrod and Amras
- Elladan and Elrohir
- Elros and Elrond
- Eluréd and Elurín
For more information, see this link to the "Ask Middle-earth" blog. Essentially, it confirms what Micah says in her excellent answer: most of the elf twins we know of are actually half-Elven. In fact, of the 12 known half-elves, 6 are twins! It is therefore likely that there is a correlation between being half-Elven and being/having twins.
There are actually a lot of twins in Middle Earth. More than seem statistically likely, but Tolkien never said if there was a specific reason for this. We meet four pairs of twins among the elves (all related), and two sets of twins among men.
Folcred and Fastred, one of the sets of twins among men, were the oldest sons of Folcwine, king of Rohan. In 2885 of the Third Age, Folcwine helped Gondor in their war with Harad. During a battle along the river Poros both Folcred and Fastred were killed, and buried on site. Their burial mound was later known as the Mound of the Twins. One of the elvish sets of twins also died together. Elured and Elurin, sons of Dior (and grandsons of Luthien) were abandoned in the forest to starve during the Kinslaying at Doriath, and were never seen again (they were only 6 years old at the time.)
In some accounts Amrod and Amras (sons of Feanor) also died together in battle, during the Kinslaying at the Havens of Sirion. However, other versions of the story give the pair a sadder fate. The two were very close, and called each other Ambarussa (“top-russet”, referring to their red hair). After Feanor and his people stole the ships from the Teleri and landed in Middle Earth, Amras slept on one of the ships. However, Feanor didn’t know this and didn’t wake him before setting fire to the ships later that night. Amrod only realized the next morning that his twin had died in the fire. Amrod was then killed towards the end of the age in the Kinslaying and the Havens...
Finally, it bears mentioning that, of the 12 peredhil (“half elves”), 6 of them are twins. Elured and Elurin, as descendants of Beren and Luthien, were half-elven. Their nephews, Elrond and Elros, were also half-elven, and Elrond’s sons Elladan and Elrohir were also half-elven. Whether this was intentional on Tolkien’s part or not is unknown, but I think it probably was, as a reflection of the duel-nature of a peredhel. Especially in the case of Elrond and Elros, where each twin literally represents one of the races of their ancestry. Elured and Elurin didn’t have a chance to make a choice about their nature, and we don’t know what Elladan and Elrohir chose. But maybe each set of twins ended up divided by fate?