This question and others like it come up a lot online.
So, lets bypass the dur-hur bullshit answers quickly. "It's all myth and fiction , so it doesn't matter" and "whatever the writer decideds."
The real answers need another pair of questions: what type of zombies and what types of vampires?
See, I'm an amateur writer ( meaning I don't generate enough income from it to wipe my ass with, but I've sold enough to say I'm a writer to some extent), a long time DM, and a mythology enthusiast. So I've stayed up nights thinking about, or debating with others this kind of question often enough to have developed a bit of a mental library on the subject. Not to mention actual books and ebooks.
Generally, zombies tend to be either magically created, or biologically spawned (be it by the living humans or nature itself). There's other types, and sub types galore. The original zombie is a creature magically created to be a servant. Here, in the real world. Now, I don't believe in magic per se, but the people that have zombies as part of their cultural history sometimes do, so real world zombies fall into the magically created category for the purposes of this question. Yeah, arguments can be made otherwise, but I'm driving this ship.
Vampires, in their real world origins are murkier because you still have to ask "which vampires"? Shit, the murk is even deeper when you realize that the *wordz vampire is old enough and has gone through enough languages to get into English that you can't even precisely point to the true origin of the word beyond something slavic or turkic, which means the nature of those vampires equally shrouded.
However, vampires, unlike zombies, are definitely dead. They die, get buried, and rise again, but most definitely not alive. See, some of the lore of zombies isn't exactly clear on that side of the bite fest. There's lore where the zombie isn't dead, just soulless, having had its soul removed, trapped, by the houngon/bokor (you pick your preferred practitioner and the spelling, I'm too tired to explain the difference) and thus is undead by virtue of not really being a person rather than being technically dead. Since known zombies had heartbeats, they explanation would fit best is suppose.
Using the most basic versions of the two, a zombie bite is no different from any human bite. But the vampire bite is still a vampire bite.
So, wtf does that mean?
It means that nothing happens. The zombie might have some nasty bacteria in its mouth, but a vampire from real world myth won't get infections. In the opposite direction, assuming the vampire bite automatically changes the target once it is killed, then there's a chance that it might successfully change the zombie into a vampire, since its body at least still has life.
Now, obviously, there were assumptions made for that. Change any given assumption, and the outcome follows suit. If real world type zombies are actually dead, they couldn't be changed. If vampires from Slavic myths were not really dead at all, who knows?
But it's a decent leaping point for thinking about such things.
Now, as any writer or DM is wont to do, lemme take you into my world for a bit.
In the shared universe I both write and DM in, there's multiple types of both. Some vampires of the most common type are made by a magical symbiotic viroid. Some of them can actually inject their symbiote into a dead body and animate it until it falls apart. Regular reinjections can extend that viability.
Some of those can cast necromancy magic instead, and raise a corpse. That corpse will stay moving as long as they put magic into it, or otherwise sustain it, or they cease to do so and it falls apart.
The main character in my self published book in that world is a necromancer of peak skill and power. His zombies will rise looking alive, and since he has the skills to do it, that's true even if the dead person is dust hundreds of years old. Just isn't easy, and it takes a shit load out of him.
In that universe, a vampire biting that kind of zombie will annoy its necromancer and may end up his next target for attention. While symbiotic vampires in that world are less vulnerable to control and manipulation via necromancy than truly undead or infernal/demonic vampires, necromancy is more than just making zombies.
However, a zombie bite is only going to annoy a vampire. In singles anyway. A swarm of the fuckers might do more damage than the vampire could heal, assuming the vampire didn't escape or otherwise prevent it. Most necromancers can't really keep a standing army of undead though. Not that it's impossible, but there's limits to how much energy you have to sustain undead. Even then, you gotta get the bodies, and put them somewhere. Ever seen a swarm of humans? Like at a concert? Imagine trying to keep thousands of zombies in your back yard. Not practical at all.
Anyways, that's my malarkey