It would be even worse if insects got it. Zombie bees, zombie ants, zombie butterflies....
There is actually a book series called The Rising that follows that premise. A military research installation opens a doorway to hell and demons are able to escape into the bodies of the recently deceased. Not to spoil the novel but, at the end of the second book in the duo, the final chapter is about the death of the last surviving animal life forms and the demons starting to overtake insects.
It was actually a fairly good series. One of the creepiest zombie works of fiction that I've ever read.
Also, as asked by another poster, zombies attacking animals other than humans depends on the author in question. In the Romero series of films (despite a brief shot of an insect being eaten) zombies are said not to pursue anything other than humans - Survival of the Dead uses this as a major plot point. The same was said of the remake of Dawn of the Dead when the dog Chips was able to walk passed zombies and survive alone in a parking garage infested by them. Nonetheless, in the 1990 remake of Night by Tom Savini a zombie can be observed eating a field mouse and, as seen in TWD, zombies did eat the horse and the swamp walker partially ate a cow before attacking Dale.