Actually our saliva may contain viruses. Its proven!
Exactly, the virus would be held in the "zombies" bodily fluids, and would infect others by entering the blood stream (Saliva into a bite wound) it's the same concept as getting rabies by a dog bite, they don't have hollow teeth like snakes to transmit their weapons, it is the saliva carrying the infection.
Also, poison is different from a virus, and venom (From a snake) is different from poison.
Believe it or not, venom is safe to drink so long as you don't have a small internal cut anywhere on the way down (Venom only takes effect when it enters the bloodstream, whereas poison can kill from skin contact, also I don't recommend drinking venom).
[Edit] By brain dead I just mean advanced functions. The zombie wouldn't be able to "think" per say, it would have the knowledge of basic movements and feeding instincts. It would see anything moving as sustenance (That is assuming that the brain functions include sight and hearing).