The Gemara (Shabbat 128b) states that animals are Muktzah. The Magid Mishnah (commentary to Rambam Hilchot Shabbat 25:25) explains that this is because animals have no utility on Shabbat and Yom Tov. Animals’ lack of purpose makes them comparable to sticks and stones, which are classified as "Muktzah Machmat Gufa", “Muktzah by its very nature.”
The Rishonim, however, debate whether an animal being used to quiet a child from crying is considered to be Muktzah. Tosafot (Shabbat 45b s.v. Hacha), Mordechai (Shabbat 316) and Hagahot Oshri (commenting on Rosh, Shabbat 3:21) cite authorities who believe that such animals are not Muktzah by virtue of the fact that they have utility. Yet Tosafot, Mordechai, Hagahot Oshri, and Rosh (cited in the responsa of Maharach Or Zarua, 82) reject these authorities because of two possible considerations. First, an animal’s ability to quiet a child from crying is simply insufficient cause to remove the creature’s status as Muktzah Machmat Gufa. Second, the rabbis classified all animals as Muktzah, regardless of whether a particular animal has utility on Shabbat and Yom Tov. This is an example of "Lo Plug Rabbanan,” a type of rabbinic legislation which was instituted for a reason, yet embraces even the cases for which the initial reason no longer applies. Shulchan Aruch (Orach Chaim 308:39) accepts the position that all animals are considered to be Muktzah without exception. Indeed, Shulchan Aruch Harav (308:78) rules stringently in this regard.
The question arises, however, whether circumstances have changed sufficiently since the time of the Rishonim to warrant a different Pesak. Modern authorities discuss animals which can be used to amuse children, but not those whose entire purpose is to entertain and provide companionship to their owners. Rav Shlomo Zalman Auerbach (cited in Shemirat Shabbat KeHilchata 27, footnote 96), in fact, raises the possibility of making this distinction, yet he rules that pets are Muktzah. Rav Moshe Feinstein (Iggerot Moshe, O.C. 4:16 and cited in "The Halachos of Muktzah" p. 7 of the Hebrew section, paragraph twenty-four) and Rav Ovadia Yosef (Teshuvot Yabia Omer, 5:26) also reject the possibility of making such a distinction. It appears that this question is contingent on one's acceptance of one of the two reasons (stated above) offered by the Rishonim) for why an animal that can be used to quiet a child from crying is Muktzah. If one adopts the position that the rabbis have deemed all animals to be Muktzah, regardless of their utility, then even household pets are to be included in this category. However, if one assumes the position that the possibility of using an animal to amuse a child is insufficient utility to remove it from being considered Muktzah, then if a pet is sufficiently useful to their owners on Shabbat and Yom Tov, it may not be Muktzah.
Rabbi Shmuel David (Sheilot U’Teshuvot MeiRosh Tzurim 38:6) concludes his discussion of this issue with a citation of the opinion of Rabbi Aharon Lichtenstein: It is proper to conduct himself in accordance with the stringent opinion in this matter, since this appears to be the opinion of Tosafot, Mordechai, Hagahot Oshri, and Rosh. Yet one need not admonish those who practice in accordance with the lenient opinion in this matter, since this issue is embroiled in a dispute amongst the Rishonim and the logic of those who rule leniently is compelling. However, even according to the stringent opinion it is reasonable to say that one may move a household pet to alleviate its suffering (Yabia Omer 5:26). This is because some authorities permit moving items which are undoubtedly Muktzah, to spare an animal from suffering (see Mishnah Berurah 305:70 and Chazon Ish 52:16). Since the question as to whether household pets are Muktzah is in dispute, there exists a Sefek Sefeika, a double doubt, which would lead one to rule leniently in this regard.
It should be emphasized, though, that one may not violate Shabbat even to save an animal's life. One may, however, ask a non-Jew to do something Assur for a Jew on Shabbat, in order to alleviate an animal's suffering. In addition, one may give a sick animal medicine on Shabbat (see, generally, Mishnah Berurah 332:5,6, and 9 and Shemirat Shabbat Kehilchata 27:54-58).