Maran (Siman 308) clearly writes that animals are Mukse on Shabbat. This applies not only to dead animals, which serve no purpose, but even live animals. In previous generations, people had chickens running around their house. The Aruch Hashulhan (Rav Yechiel Michel Epstein of Nevarduk, 1829-1908) made a point of warning women not to handle their chickens on Shabbat. Accordingly, if a cat is sitting on the chair, one may not physically remove it to sit down.
The Poskim discuss whether animals which serve a decorative purpose, such as a fish in an aquarium, are Mukse. Do the fish have a status of a household furnishing and may the aquarium be handled? Rav Moshe Feinstein (Russia-New York, 1895-1986) in Iggerot Moshe (OC 4:16) rules that fish are Mukse, and therefore the aquarium becomes Mukse as a "Basis." This is also the opinion of Hacham Ovadia in Yabia Omer (5:26) and in Halichot Olam (Vol. 3, Parashat Vayigash 1), and Rabbi Moshe Halevy (Israel, 1961-2001) in his Menuhat Ahaba (1:12:48). This also applies to a parrot which kids play with. Both the bird and its cage are Mukse. Designating live animals as a decoration or play item will not help.
However, there is a minority opinion that holds that although the fish are Mukse, their tank is not Mukse and may be moved, assuming that there is no issue of electrical wiring and so forth. They reason that the tank is a household decoration and does not become Mukse as a "Basis," because it's all one entity. This is the opinion of Rabbi Haim Pinhas Sheinberg (Jerusalem, 1910-2012) in his Kuntres HaMukse printed at the end of Shalmeh Yehuda, and the Shemirat Shabbat K'hilhata (Ch 27, note 97) in the name of Rav Shlomo Zalman Auerbach (Jerusalem, 1910-1995).
SUMMARY
All animals are Mukse, including parrots, fish and puppies used for decoration or play. The fish tank and birdcage are also Mukse.
