One of my colleagues recently asked this question. Through a series of detailed logic-gates, we can conclude the following two scenarios:
In much zombie lore, zombies don’t like to eat (long) dead things or even diseased humans (cf. World War Z, the movie). Therefore: vampires win because they’re immune.
However, if the cause of zombie-ism was a “virus” that effected any dead body, not just one bitten by zombie, (cf. The Walking Dead, the TV show), then the vampires may be in trouble as they’re technically “dead” and would seem to be contagious to the zombie-flu in their dead state.
Addendum – food supply problem for vampires
In scenario one (vampires win) a problem presents itself. What if all living humans were turned to zombies and there were no more living humans?
Vampires must eat living humans, so their food supply would be eliminated. Some vampire-lore seems to indicate that vampires can live off rat blood and other creatures, while others suggest a synthetic blood can be invented (cf. True Blood, the TV show), while others seem to indicate that vampires can hibernate for centuries (cf. the Underworld movies) at least holding off death (though it seems to require a great deal of blood to setup). So, if the food supply was eliminated, problems arise for vampires, assuming they can’t drink whatever “zombie blood” is.
Minus the synthetic blood deus ex machina (and rats), in these situations, the vampires would have to clone humans (cf. Hemlock Grove) or find living humans (astronauts trapped in space?) then breed them. It’s dicey (because science), but vampires still win.