Just as a Jew is commanded to rest on Shabbat, so too, he is commanded to allow his animals to rest. There are two commandments that address this issue, one positive (aseh), as the Torah states (Shemot 23:12): “Six days you shall do your work, but on the seventh day you shall cease, so that your ox and your donkey may rest, and the son of your maidservant and the stranger may be refreshed.” And one negative (lo ta’aseh), as the Torah states (Shemot 20:10): “But the seventh day is a Shabbat of the Lord your God; you shall not do any melakha – you, your son or daughter, your male or female slave, or your beast, or the stranger who is within your settlements.” The mitzva of allowing animals to rest is not one of the 39 melakhot but a mitzva in its own right; its violation is not punishable by death or lashes (Shabbat 154a; MT 20:1-2).
Even though the Torah in those verses mentions beasts, oxen, and donkeys, the prohibition applies to all animals, including birds and fish (MB 305:1). Therefore, one may not release carrier pigeons on Shabbat, and one may not use trained dolphins to pull a boat. The Torah specifies the ox and the donkey because they are the beasts of burden most commonly used for plowing and carrying heavy loads.
One who causes an animal carrying a load to move from one place to another on Shabbat, whether by striking, pulling, or ordering it, transgresses Torah law. This prohibition is referred to as Meĥamer. Even if the animal belongs to a non-Jew or is ownerless, one may not cause it to walk with a burden. If one causes his own animal to do so, then in addition to transgressing the lo ta’aseh against doing work, he has also violated the aseh of allowing animals to rest (SA 266:1-2; MB ad loc. 7-8). The latter commandment applies on Yom Kippur as well (there is disagreement about whether it applies on Yom Tov; see MB 246:19).
A Jew may not rent his animal to a non-Jew who will use it to do work like plowing or carrying burdens in the public domain on Shabbat. If a Jew rents out an animal with the understanding that it will be returned before Shabbat, but it is not, he should declare the animal ownerless before Shabbat to avoid violating Torah law (SA 246:3). If a Jew and a non-Jew are co-owners of an animal, the Jew may not permit the non-Jew to work the animal on Shabbat. However, if when they first bought the animal they specified that the non-Jew would be the sole owner on Shabbat, while the Jew would be the sole owner on a different day of the week, then the non-Jew may work the animal on Shabbat, because on Shabbat it is his alone (SA 246:5).
A Jew may allow a non-Jew to ride his horse or other animal on Shabbat, in accordance with the principle that “a living being carries itself” (he-ĥai nosei et atzmo). Therefore, the non-Jew who rides the animal is not considered a burden (nor are the clothes he wears, as they are secondary to his body). However, the Sages enacted that a Jew may not use an animal at all. He may not lean on it, place an object on it, or sit in a wagon hitched to it, even if a non-Jew is driving the wagon for his own purposes (SA 305:18). The reason for this enactment is to avoid burdening the animal on Shabbat (y. Beitza 5:2). Another reason is that while riding an animal, one may come to break off a tree branch in order to goad the animal, thus transgressing Kotzer (Beitza 36b).
A Jew may not take another Jew’s animal outside of teĥum Shabbat. This teĥum is based on the teĥum of the animal’s owner (see below, 30:3). If the owner entrusted the animal to a shepherd, whether Jew or non-Jew, the teĥum is based on that of the shepherd (SA 397:3-5). The prohibition is only for the animal to leave the teĥum at the initiative of its owner; it is not prohibited for the animal to leave of its own volition or for a non-Jewish shepherd to take it outside the teĥum (Rema 305:23; MB ad loc. 79).