I wonder why this question took so much time to be addressed. The topic has been discussed at many a cat-loving shabbos table. I know many cats that have been cajoled into saving Shabbat for their humans. (The same would apply to dogs, rabbits, gerbils, and ferrets; snakes remain a tzarech iyyun.) There is still the question of training the animal to be your shabbos-cat. What about amirah le’kitty combined with another rabbinic prohibition?
From Tzomet’s Shabbat B’ Shabbato
Volume 1504: Shemot 18 Tevet 5774 21/12/2013
Halacha From The Source
Getting a Pet Cat to Turn on the Lights on Shabbat / Rabbi Yosef Tzvi Rimon, Rabbi of Southern Alon Shevut and a teacher in Yeshivat Har Etzion
We have been asked if it is permissible for somebody to hold a pet cat near a switch so that it will turn on a light that was left off before Shabbat by mistake.
Having a Pet perform Prohibited Labor
We differentiate between different ways of accomplishing the goal of having the pet turn on a light.
(1) Pushing the pet against the light switch – In this case, the action is done by the human being, and this is forbidden. Even though this might be viewed as performing the labor “beshinui” – in a modified form – since it is being done in a way that is not usual behavior during the week, but in this case even labor performed beshinui is forbidden (Shabbat 92a).
(2) Holding the cat in the air in such a way that in a natural motion it will almost definitely move its feet and kick the switch. The question in this case is who is actually performing the action.
(a) If we say that the person is performing the action while making use of the natural habits of the cat, this might well be similar to “zoreh,” one who separates the straw from the grain, with the help of a wind (Bava Kama 60a) or to one who puts a leech on a human being in order to suck out the blood (Magen Avraham 328:53; Even Ha’Ozer, ibid). Both of these actions are forbidden.
(b) If, on the other hand, we define that action is the result of both the cat and the person acting together, this would be a violation of the labor of “mechamer” – leading a donkey – which prohibits a person from doing something together with his animal (Shemot 20:9; Shabbat 153b-154a; Shulchan Aruch 266:2.)
(c) Even if we define that the cat performs the action alone in a way that corresponds with the will of the person, this is evidently included in the prohibition of giving work to an animal in your possession (Shemot 20:9; Avoda Aats 15b; Sulchan Aruch 246:3).
According to all three of the above definitions, this action is prohibited on Shabbat.
(3) However, if we hold the cat in a stable position a short distance from the switch in such a way that it is not forced by its nature to touch it, and the cat itself “decides” to stretch out its foot and play with the switch until the light is turned on – this might well be permitted in a case of great need. It would be similar to the case allowed in the Talmud of taking a small child “for a walk” near an item that has fallen down (where there is no “eiruv” to allow carrying), so that the child will pick the item up by himself and bring it home (Yevamot 113b-114a). The Rashba derives from this that one is permitted to put a young baby “near” a prohibited item that he needs “so that he will put out his hand and eat it” (in practice the use of a child is more complicated than indicated here, but we will not discuss this further).
Evidently the same principle applies to holding an animal close to a forbidden item so that it will perform labor, since the obligation for the animal to rest on Shabbat is derived from the same verse (Rashi, Ramban, ibid; Rashba Shabbat 153b; Chatam Sofer volume 1 (Orach Chaim), 83; Responsa Achiezer 3:83; Orchot Shabbat 24:7-8, note 2, and 31:6-10).
Muktzeh: Recent rabbis do not agree whether a pet can be moved on Shabbat or it must be considered muktzeh and therefore not be touched. One who acts in a lenient way has a valid opinion on which he can base his action.
Using a cat or other pets to turn on the light: If one physically pushes the pet onto the switch or if it is placed in such a way that by its nature it is almost certain that it will push the switch, this should evidently be prohibited. However, if the pet is held in a stable way a distance from the switch, such that there is no certainty that it will push the switch – this can evidently be permitted in a case of great need. Full Version Here
h/t Tomer Persico