I wrote a huge answer about this before in a very similar question, but I can’t find it anymore. And anyways, I have more class (not much, but some) then to copy an old answer and just paste it again. I’m not Vega from Street Fighter. My answer won’t be as big this time though, but I’ll try to get to the main points, as decreed by traditional horror themes, or at least what I personally grasp of them.
Zombie
The zombie virus only spreads to the living. It won’t ever be interested in a vampire, because a vampire is undead, and the zombie probably wouldn’t even register it. But if it did bite a vampire, nothing would happen, because again, the vampire is undead, and the zombie virus is only applicable to the living, and usually humans.
If it bit a werewolf though…well, it wouldn’t have time to bite the werewolf. The werewolf would kick its ass so fast. That’s what werewolves do. They destroy all that moves.
But if it did bite a werewolf…it wouldn’t be affected. The werewolf is already afflicted by lycanthrope, which, in the spectrum of the undead, (although a werewolf technically isn’t even fuckin undead) is a curse much stronger than the zombie virus. If the werewolf reverted back to human though, and THEN was bit by a zombie, it may or may not turn into one. If lycanthrope is more a curse than an actual disease, then the virus may be invalid, but if lycanthrope is a disease, it would be overwhelmed by the zombie virus which affects the brain at its core, and would over wash whatever exaggerated version of rabies that lycanhropy mythologized itself from.
I’d sure love to see a zombie werewolf though.
Vampire
The vampire is the strongest undead thing that exists, and it is cursed by God. The vampire is more apt to control an army of zombies just by looking at them for not even a second, and a werewolf is the vampire’s bitch. (I love Underworld, but still, fuck those movies lol)
Only humans can be turned into vampires, (and pillows, according to some 18th century literature I own, although it wasn’t actually a pillow, but a huge parasite living inside a pillow…pillows! :D ) and a zombie is no longer human. Nothing would happen if a vampire bit a zombie.
The werewolf is confusing though. It is human most of the time. I’m assuming that being turned to the vampire side might actually be a good cure to lycanthrope, if you don’t mind eternal damnation. A vampire biting a werewolf when it’s in wolf shape though…I think it would work, only the werewolf would turn into a vampire after it reverts back to human form, or perhaps the curse of the vampire is strong enough to destroy the lycanthrope right off the bat. (again, disease VS curse) The werewolf is still living, human or not. And technically, it is a human, so if anything, the curse may be withheld until the other curse fucks off.
Werewolf
If a werewolf bites something…well, for lycanthrope to work, you need an immune system. You know, so it can be all affected and shit. Zombies are dead and have none, same for vampires. A zombie bit by a werewolf would be torn to shreds, and a vampire would release a small laugh and make it its slave, or send it to hell, depending on if he’s having a good day or not.
In the end, the zombie wins the rock paper scissor match, because it just don’t give a fuck.