As we will see later on (23:5), anything that has no practical use on Shabbat is muktzeh and may not be carried. Animals are included in this category and thus may not be carried on Shabbat. If, in order to prevent them from being hurt, it is necessary to move them, the Sages permitted pulling them but not picking them up (SA 308:39-40). At first glance, this would seem to pertain to house pets like cats and dogs, and indeed, this is the position of Yalkut Yosef (vol. 2, p. 383) and Orĥot Shabbat (19:124). However, it seems more reasonable to assume that muktzeh pertains only to animals with which one does not normally play. Pets whose owners play with them and pick them up all week long would not be muktzeh, and their owners may thus touch them and pick them up on Shabbat. Similarly, seeing-eye dogs are not muktzeh (Igrot Moshe, OĤ 5:22:21; Shulĥan Shlomo 308:74). Although some are stringent, muktzeh is a rabbinic prohibition so the halakha follows those who are lenient.
A fish that jumped out of an aquarium and will presumably stay alive if it is returned may be put back even though it is muktzeh. This is because, when necessary, and when there is no other solution, one may move a muktzeh object to prevent tza’ar ba’alei ĥayim. Even though some are stringent in this case, one may rely on the lenient approach (See SA 305:19; MB ad loc. 70; SSK 27:28).
If a fish in an aquarium dies, and it is possible that if it is left in the aquarium other fish will catch whatever terminal disease it had, then even though the dead fish is muktzeh, it may be removed from the tank to save the remaining fish. If there is a dog or cat in the vicinity, the fish should be fed to them (see SSK 27:29).