I've never heard of an animal removing a ball from play entirely before. The PBUC Umpire Manual does suggest what should happen in the slightly more likely case of striking an animal.
7.21 Ball strikes bird or animal
If a batted or thrown ball strikes a bird in flight or other animal on the playing field, consider the ball alive and in play the same as if it had not touched the bird or animal. If a pitched ball strikes a bird in flight or other animal on the playing field the pitch is nullified and play shall be resumed with the previous count.
An "ask the umpire" piece on MLB notes what could happen for cases that are not covered by the rules (such as an animal removing the ball from play).
When a situation is not covered, Rule 9.01(c) comes into play. That rule gives the umpire authority to rule on any point not specifically covered in the Rules. In such instances the umpire is instructed to use "common sense and fair play."
I suspect that any ruling in such an unusual situation would depend greatly on the specifics.