In The Silmarillion, and in the appendixes to The Lord of the Ring, JRR Tolkien writes that the Eldar, or Elves, are like humans, but that they are immortal, and can not be killed by any means other than grief or wounds. Explicitly excluded from the list of things that can harm elves is sickness of any type.

However, contradicting this is at least two allusions in Tolkien's canonical works that mention Elves being poisoned. The first is the story of Celebrian, wife of Elrond, who was captured, poisoned and tormented by Orcs in the Misty Mountains. The other reference is the slightly more trivial example of the Elven butler in The Hobbit, who has his keys stolen from him by Bilbo while he was in a drunken stupor. A drunken stupor could definitly be considered a form of poisoning.

Although the Hobbit was not at the time it was written about Tolkien's canonical elves, it later became a canonical book. As such, we have no reason to believe that elves cannot become intoxicated with the effects of alcohol. They can also be poisoned seriously.

However, there is no evidence of Elves being poisoned enough that they could be killed, as that would be against the fact of Elvish immortality. Whether or not Elves would be able to catch a contagious disease that works by producing toxins in the body is also not addressed, but even assuming it is possible assumes that the mechanism of disease in Middle Earth is the same as it is on our earth.