To save a human life, it is permitted – even halachically obligatory – to violate Shabbat. In fact, one should be sufficiently familiar with the laws so that he knows what to do without asking. That is not the case with saving the life of an animal. Nevertheless, the principle of avoiding the suffering of animals does come into play.

R. Yehuda said in Rav’s name: If an animal falls into a water canal, one brings pillows and bedding and places [them] under it [despite the rabbinic law of mevatel k’li meiheichano[1]], and if it ascends [out of the canal], it ascends . . . [The avoidance of] suffering of animals is a biblical [law], so the biblical law comes and supersedes the [law] of the Rabbis.[2]

Similarly, one may ask a non-Jew to milk a cow on Shabbat so that the accumulation of milk will not cause it pain.[3] There is significant disagreement as to which rabbinic laws can be permitted in order to prevent pain to animals.[4] Violation of Torah laws, such as driving a car, is certainly forbidden.
Notice, however, that leniency applies only to steps on Shabbat to prevent actual pain to the animal. (There is discussion whether or not the aforementioned leniency applies to saving an animal’s life if its death would not be painful.)[5] That is not the case if one has to violate Shabbat to retrieve the animal from a hospital or shelter. The animal will not be in pain if it remains until Monday. Additional expense for the hospital stay would not be cause to waive even rabbinic laws.