I'm gonna say they both remain unchanged, but the Vampire is now a carrier of the zombie virus (if it's that type of zombie).
- Vampires turn humans into vampires. Zombies are no longer human. A vampire biting a dog does not make a vampire dog.
- I'm more convinced that a vampire could become a zombie (any mammal can become a zombie), but I'm going to rule that vampires are already undead, so the process to make them turn undead has no effect. That said, if the zombie has a virus, it would still populare the vampire's body. Sort of making them an asymptomatic carrier for it, I guess.