I would base my response on the creature in the lake outside Moria. It focused on Frodo because he was the ring bearer. There’s a lot we don’t know about that creature, but I have to assume that any force able to manipulate it would also be able to manipulate a dog or a bird.
There’s also a scene where the group is attacked by wolves, though without specific mention of the Ring.
Of course, we do have another “out” on this issue. If Gandalf has possession of a bird and that bird has possession of the ring, then Gandalf has essentially taken possession of the ring himself. (In the legal sense, if not in a physical sense). It could very well be that this is still a sufficiently strong connection to the ring that Gandalf would find himself unable to compel the bird to destroy the ring.
This possession-by-proxy idea would give us an answer we could apply to a robot or AI. C3PO might not be corrupted directly, but his owner would be unable to order the destruction of the ring.
BigT’s answer is also one that I find convincing. Part of the ring’s function is to take itself back to Sauron. In fact, I’d argue that Frodo’s quest to destroy the ring is an example of how subtle the ring is. Frodo is strong enough to refuse to wield power himself and to refuse to bring it to Sauron directly. And yet he volunteers himself to carry the ring to right next to Sauron, justifying it with the idea that he can destroy it. Once there, he finds himself unable to destroy it and the quest would have played directly into Sauron’s hands except that Gollum takes the ring and destroys it by accident.
So… a snake isn’t smart enough to need all that rationalization, but could be compelled in a similar manner. Someone observing the snake might not see a direct line of travel, but might still find that every day the snake had wandered a little bit farther south than the day before.