Wait wait wait, first of all what sort of zombies are we talking here?
The dynamics of the zombie need to be addressed before any assumption can be made.
A night of the living dead / The Walking Dead zombie, which is your typical slouch-around slow zombie, would stand absolutely no chance against a vampire, purely on the basis they lack the required motor-neurone skills to even get near a vampire - which is superior in every way. That, combined with the fact that the zombie effect only takes place once the host body has died. Since a vampire is already technically dead no effect would take place. That is unless the vampire is then slain, in which would it rise again as an undead-vampire. The effects this would have on a vampire-human contact though is still up for debate as to whether the infection/virus is carried along to the human as well as the vampire genetics.
This would also apply to your standard runner zombies like Resident evil, as whilst these are faster and more lethal - again biting an undead creature would have no effect.
28 days later zombies? Well these fellas are infected with Rage, so they are faster and stronger than a usual zomb, so one bite or scratch on Mr Count Drac and he'd become infected too, turning him into some pimped-out super-mad vampire! Now if this were to happen we'd have a bad day, as not only would a vampire bite transfer to humans to make us vampires, but the rage would too.
So these basic parameters should be looked at matthew$ before we continue this discussion any further.
What if a ragey zombie bites a slow zombie that then bites a crappy vampire from black and white films?. Does it become a fast angry bat on a string that might bite you in your sleep and make you black and white?
also what if your born a vampire like blade surely his human side would let the infection take place.Furthur more to answer dodge-gnome im thinking these type of zombies would surely overcome a vampire