Not only is it prohibited to put animals to work on Shabbat, but one may generally not even move an animal, as they are classified as muktze. It is also prohibited on Shabbat to feed animals that are not one’s direct responsibility.

One must provide food and water to animals that are in his possession and which depend on him, such as cows and chickens. It is prohibited to give food and water to animals that can find food on their own, such as doves from a dovecote, or bees. Although feeding them is not a prohibited labor by Torah law, it is nevertheless rabbinically forbidden.

A hungry animal which has not found food on its own may be fed on Shabbat, even if it is not one’s responsibility to feed it, in order to avoid causing suffering to animals.

One who is accustomed to throwing leftover food into a specific area may do so on Shabbat as well, even if the food attracts animals that are not owned by him or are not his responsibility.

It is prohibited to move animals on Shabbat, as they are defined as muktze. With regard to pets that one usually plays with, there are differences of opinion among halakhic authorities as to whether or not it is permitted to pick them up and play with them on Shabbat.

In a situation that involves the suffering of living beings, such as animals that are stuck in a cage or in the burning sun, one may carry them in order to spare them pain. If a fish died in a fish tank and there is a concern that the other fish might die if the dead one is left in the water, it is permitted to remove it from the fish tank on Shabbat and to dispose of it.

Further reading: For more on the prohibition of causing animals to suffer, see p. 613; A Concise Guide to the Sages, p. 435; A Concise Guide to Mahshava, p. 122.