What, if anything, happens to a vampire who is prevented from drinking blood?

Solution 1:

Not much, really. In fact, this is a pretty common tactic with vampires. The cool thing is, they'll still follow orders and be productive dwarves, so they're very useful as an immortal bookkeeper and/or lever puller. Just seal them away in a room with your levers, and they'll always be available.

They don't need food or drink, but you will want to keep clothing available to them, as they can still get in a bad mood. A couple methods of doing so are to either make an airlock between their room and the fortress, or have a drop shaft above their room where dwarves can drop in new clothing.

This will also prevent your fortress from ever falling. Even if the rest of your dwarves are killed, that one immortal bookkeeper will keep it going until more migrants can arrive.