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Is it permissible to move animals on Shabbat, such as to lift one's pet dog, cat or bird to play with it, or to move one's animal to use the space it occupies, such as if one's pet is sitting on his chair or couch?
Animals have the status on Shabbat of "Muktzeh Machamat Gufo," the strictest category of Muktzeh. Therefore, it is forbidden on Shabbat to lift an animal on Shabbat either to play with it or to use the space it occupies. Furthermore, if a bird or some other animal is in a cage, the cage, which serves as a "base" for the Muktzeh item, becomes Muktzeh itself, and thus one may not move the cage, even if he needs the space.
Likewise, Chacham Ovadia Yosef ruled that an aquarium or fish tank containing fish has the status of Muktzeh Machamat Gufo and may not be moved on Shabbat. Even if the tank occupies counter space that one needs, it is forbidden to move the cage on Shabbat.
It is permissible to feed animals, such as by placing food in front of them or pouring fish food into a tank.
Summary: It is forbidden on Shabbat to move an animal or a cage containing an animal including a fish tank for any reason, even to use the space. One may feed animals and fish on Shabbat.