Vampires are undead, and as such are no longer living--can't taste or digest food, can't reproduce, and doomed to spend the rest of their time in a perpetual state of decay. But this is fine. I mean, you're no longer aging, either!
The reason the state of decay isn't really noticed is because vampires are also constantly regenerating from anything which may need healing. This is why many vampires like to sleep in the cold, where the decay is slowed, and the need for healing is slowed, and that energetic buzz which goes along with it is less noticeable and sleep more restful. Healing works the same as it would for mortals: skin and muscle tissues heal the fastest, with broken bones taking perhaps longer to heal--it's just that it's much faster, and more efficient too (no scarring). Healing costs energy, which is restored through rest and blood replenishment. Healing only works so long as there is energy to heal, which means if something makes you sick, the decay will begin to take hold. Aside from mere lack of sleep or blood, some things commonly known to foil a vampire's healing are silver in the bloodstream, a wooden stake too close to the heart, and especially the sun.
Any amount of sun is too much sun. While they may not burst into flames, the sun is essentially toxic and will halt the vampiric regenerative ability almost immediately. It hurts and makes them short of breath, and would result in a feverish, hallucinogenic, nearly comatose state in under an hour--with the time-speed that is Minecraft, this means about a minute. Think whole-body organ failure, bones become brittle, that sort of thing. A 'wither' effect would fit here nicely.
This ain't Skyrim, folks. While shielding the eyes and the face may help with the blinding pain, it does not prevent damage for the traditional vampire.