I don't think the virus would even be transmitted. The vampire concept has not been as thoroughly developed as the zombie one...
And we also need to examine various cases of "zombie viruses":
Depending on whether the zombie virus can effectively transcend the species barrier (i.e. whether it can infect animals with the same or similar effects as human infection), the results can vary greatly
- If the zombie virus is restricted to human infection, then the vampires are obviously immune, for two important reasons:
1) Vampires have the accelerated healing factor. Meaning that nearly any injury, aside from some specific kinds of injuries (depending on the lore), leaves the vampire near-unaffected, due to the rate at which it is healed.
Given this healing factor, there is no reason to consider the chance that it doesn't work with diseases. An advanced healing factor such as this, is almost definite to be able to combat infections from viruses, bacterai, parasites, anything most likely.
2)Depending on the lore, vampires are sometimes not even considered human, so they may be well out of the virus' reach.
The latter case however is more interesting and the one more common in zombie culture: that the zombie virus can transcend the species barrier, effectively turning both humans and animals (most commonly dogs) in zombies.
In that case, it is not entirely clear how a vampire will be affected, if at all. However, assuming that it can be infected:
1) the healing factor comes into play again. It is important to note that it differs from just a highly advanced immune system, due to the rapid rate at which injuries are healed, be they fractured bones, smashed heads or even more gruesome and extreme in their nature
2) Drinking human blood significantly affects the vampires in almost every way. Often, vampires are depicted as in need of a constant supply of human blood, which makes it the equivalent of food for humans. That said, it is highly possible that a constant consumption of healthy human blood can regulate the effects of such a virus,, even if the healing factor cannot fight it.
3) Vampires however cannot drink infected human blood, sometimes just with a specific range of diseases, but mostly with any disease. Given that they cannot consume such blood, the effects such a consumption (if forced for example) would have on a vampire are unclear.
-- However one thing is clear in such cases:
if infection from the disease present in the blood can affect the vampire "consuming" it, then, in that case, the healing factor may not be able to actually combat any disease. As such, in that (admittedly specific) case, the vampires' accelerated healing may not be able to prevent the infection or slow the progress of such a virus.
In conclusion: this question breaks down into countless different cases. Many more conditions must be established for it to even be able to be discussed.