Gemara

The Mishnah forbids selling a work animal to a Nochri. This is a decree lest one rent or lend an animal to him, and he will make it work on Shabbos.

Objection: One who rents or borrows an animal acquires it (for that period of time. It is not considered the Yisrael's animal!)

Answer (R. Yitzchak brei d'Rav Mesharshiya): No, rental does not acquire!

(Mishnah): If a Yisrael rents a cow from a Kohen, he may feed it vetch (fodder that people can eat in pressed circumstances) of Terumah. If a Kohen rents a cow from a Yisrael, he may not feed it vetch of Terumah.

If rental acquired, the Kohen would own it, and he could feed it Terumah!

20b (Mishnah): One may not rent a house to a Nochri to live in, for he would bring in his idolatry, and this is forbidden - "v'Lo Savi So'evah El Beisecha".

Bava Metzia 56b - Question (R. Zeira): Does Ona'ah apply to rental?

The verse of Ona'ah discusses a sale, not rental;

Or, perhaps we do not distinguish!

Answer (Abaye): The verse does not specify a permanent sale. Rental is a temporary sale (so Ona'ah applies to it).

Kesuvos 101b (Mishnah): Levi married Leah and agreed to feed her daughter Dinah for five years. Leah was divorced, and made the same deal with David. Both men must feed her. They cannot say 'we will feed her together.' Rather, one feeds her, and the other pays her the monetary equivalent.

103a: Reuven rented a mill to Shimon. In lieu of paying, Shimon would grind Reuven's grain. Reuven waxed rich, and bought a mill and a donkey to grind. He wanted to grind for himself, and that Shimon will pay money. Shimon refused.

(Ravina): This is like our Mishnah. One man feeds her, and one pays money.

Rejection (Rav Avira): The cases are different! A girl does not need two rations of food. Shimon can tell Reuven, grind and sell to others, and I will grind for you! If Shimon can find people who will pay him to grind for them, we force him to do so.

Rishonim

The Rif and Rosh (Bava Metzia 32a and 4:21) bring Abaye's answer that rental is a temporary sale, therefore Ona'ah applies to rental (of Metaltelim - Rosh).

Nimukei Yosef (DH Gemara): The Ran says that rental is a temporary sale for usage, but the item itself still belongs to the owner. Therefore, one may not rent to a Nochri for residence, for he will bring idolatry into the Yisrael's house, and a Kohen who rented a Yisrael's animal may not feed it Terumah.

Rambam (Hilchos Mechirah 13:17): Ona'ah (overcharging) applies to a rental of a Keli or animal, for rental is a sale for the day.

Rambam (29:8): If a minor without an overseer understands commerce, he can buy or sell Metaltelim.

Ra'avad: If he rented the place of the Peros he did not acquire, for rental is a temporary sale (and a minor cannot buy land).

Question: In Bava Metzia (56b), we concluded that a rental is a temporary sale. In Avodah Zarah, we concluded that it is not!

Answer #1 (Tosfos Bava Metzia 56b DH v'Hai): Regarding Ona'ah it says "Memkar" superfluously, to include a rental.

Answer #2 (Tosfos R. Elchanan Avodah Zarah 21a DH Af): A rental is like a sale only according to the opinion that Kinyan Peros (rights of usage) is like Kinyan ha'Guf (owning the matter itself).

Rambam (Hilchos Sechirus 6:5): One's yard acquires for him, even if it is rented out.

Question (Beis Yosef CM 313 DH veha'Rambam): Rental is like a sale for that period of time!

Answer #1 (Shach CM 313:1): Tosfos says that rental is like a sale only regarding Ona'ah. The Bach (Sof DH ha'Socher) says that mid'Rabanan, it is like a sale in other respects. I do not know his source.

Gra (CM 313:5): The Rosh (Teshuvah 1:1) explicitly says that rental is like a sale to acquire. All the Meforshim are unlike the Rambam.

Rambam (7:1): Just like one can stipulate about a sale, he can stipulate about rental, for it is a temporary sale. One can rent out something only if he can sell it.

Question: Tosfos proved that rental is a temporary sale only regarding Ona'ah! The Rambam (Hilchos Shabbos 20:3) forbids renting animals to Nochrim, lest the Yisrael's animal work on Shabbos (for rental is not a sale)!

Answer (and Answer #2 to the previous question - Lechem Mishneh): The Nimukei Yosef taught that rental is a sale regarding usage of the item. The item itself still belongs to the owner (and acquires for him).

Rosh (Bava Kama 5:10): One who rented his field to another cannot forbid it through a vow. Rental is a temporary sale; he sold benefit from the field.

Rosh (Kesuvos 12:5): Tosfos says that the renter can still grind and not pay, for the owner will not lose, for he can grind for others. This is wrong! Rental is a temporary sale. If someone must lose, the owner cannot retract. He fulfills the contract. Who told him to buy a mill?! The Gemara said that the renter can say 'grind for others', but the Halachah does not depend on this.

Poskim

Shulchan Aruch (CM 227:35): Ona'ah applies to a rental of a Keli or animal, for rental is a sale for the day.

Shulchan Aruch (315:2): One can stipulate about a sale, just like about rental. One can rent out something only if he can sell it, or if he owns the Peros.

Rema (334:1): If someone rented a house to live in and died, the heir pays only for the time he lived there. The owner is like a worker, who (is not paid in a case of Ones, for he) should have stipulated. However, some argue. Therefore, if he already received the rental, he need not return it.

Source (Beis Yosef 312 DH Kasav bi'Teshuvas ha'Rashba): Reuven rented a house to Shimon for a year, and Shimon died after two months. Shimon's son did not need the house, and wanted to pay for only the two months. The Rashba (1028) ruled that a rental is a sale for that time, whether or not he lives there. Tosfos (Bava Basra 51b DH b'Matanah) agrees; the Mordechai (Bava Metzia 345) disagrees.

Rebuttal (Bach DH Kasav Beis Yosef): Tosfos says that a rental is a sale only regarding Ona'ah. For other matters, mid'Oraisa it is not a sale. Therefore, Reuven cannot take from the heirs; death is the ultimate Ones. Tosfos in Bava Basra said only that if the heirs want the house, Reuven cannot evict them before the year.

Machaneh Efrayim (Sechirus 5): The Mordechai himself says that if Levi hired a teacher and the teacher got sick, Levi must pay him, for Sechirus is like a sale! There, Levi's bad Mazel caused the teacher he hired to get sick. Here, we cannot attribute the renter's death to bad Mazel of the house.

Shach (2): Even if rental is a sale for the day, each day is a separate sale, so the heirs need not pay for days after death. And even if it is like a sale for a year, Ones can nullify a sale. However, if he already paid the rental, he consented that the owner keep it even if there will be an Ones.

Tosfos R. Elchanan (ibid.): We hold like Reish Lakish, that Kinyan Peros Lav k'Kinyan ha'Guf. Even the other opinion distinguishes between a rental and a temporary sale. A renter is obligated to guard the animal; a buyer is not. If a landlord's house fell, he can tell the renter 'you are no better than I am (I want my house back)', but a seller cannot say this.

Gilyon Maharsha: Bava Kama 68a says that a thief does not pay four or five for a sale of 30 days. (We learn from "he will slaughter or sell it" - just like slaughter is irreversible, also a sale.) Mutzal me'Esh says that rental from Hekdesh is like Hotza'ah (removal of property) from the Reshus of Hekdesh. Living for free on Hekdesh land is like benefit without Hotza'ah.