The vampire would most likely not turn into a zombie, but that depends on how the zombies work in whatever universe.

Traditionally, zombies are immune from death magic.

I would expect nothing would happen.

Vampire might be confused and not like the taste. And if the the vampire were experienced, may even know that the zombie is immune and pointless, either from looking or smelling, or even sensing supernaturally.

At most, maybe the vampire, if desperate, might be able to "eat" from zombies, though it would probably be disgusting and purely an act of desperation.

Either way, the he vampire is 100% the superior demon/being. Vampires are aware, zombies are kinda just trash minions. Unless you're talking resident evil/racoon corp/purely bio/scifi, then all bets are off, but that's hard to truly consider a zombie, and more of an offshoot novelty nu-type.