The Vulcan Nerve pinch seems to be effective on a wide range of humanoid species (Human, other Vulcans, Romulans, etc) as well as certain animal species, notably a horse in the Star Trek V film.
Spock knocks out a giant panther-like creature in the "When Worlds Collide" webcomic. Interestingly, because it's a Robert Orci creation, it's actually a higher canon level than the animated series (see below)
As far as other animals are concerned, we see the Nerve Pinch being used against an alien creature known as a Le-Matya in the Star Trek animated series.
For the record, I'm unaware of the Nerve Pinch ever being used on anything as small as a cat or a dog but there's certainly no reason to assume that it wouldn't work.