Elves' immunity to disease does seem to me to be somewhat problematic.
We know that Elves can be affected, and killed, by physical damage. All diseases work by inflicting some kind of physical damage on the sufferer. It might be argued that a distinction may be made between external trauma (such as that inflicted by a weapon) and internal trauma (such as that inflicted by disease), yet Elves are affected, and can be killed, by poison (cf Celebrian and Aredhel), which, like disease, works by inflicting internal trauma.
The only explanation that I can come up with is that Elves had a highly developed immune system. Their vulnerability to poison, however, mitigates against this too. And if one were to speculate that the poison needed to render damage to an Elf would have to be particularly potent, would that not open them up to the possibility of being affected by a particularly virulent disease?
On different note, I have always thought it rather ironic that Elves, being the only race that do not seem to indulge in the pipeweed habit (cf Legolas' views on the habit at Isengard), are also the only race that would be immune to its potentially harmful consequences. Then again, there seem to be no recorded instances of pipeweed-related deaths in Middle-earth and the healthy lifespan of most Hobbits would seem to suggest that pipweed in Middle-earth has somewhat less potentially harmful consequences than its real-life counterparts.
