Gemara

15a: One who rents or borrows acquires it (for that period of time).

Rejection (R. Yitzchak brei d'Rav Mesharshiya): Rental does not acquire!

(Mishnah): If a Yisrael rents a cow from a Kohen, he may feed it vetch or Terumah. (Vetch (horsebean) is normally fed to animals, but people can eat it.) If a Kohen rents a cow from a Yisrael, even though the Kohen must feed it, he may not feed it vetch of Terumah.

If rental acquired, the Kohen would own the cow, and he would be allowed to feed it vetch of Terumah!

20b (Mishnah - R. Meir): We may not (sell or) rent houses to Nochrim in Eretz Yisrael. In Surya we may rent. In Chutz la'Aretz we may (even) sell.

R. Yosi says, in Eretz Yisrael we may rent (but not sell) houses. In Surya and Chutz la'Aretz we may sell.

Even where one may rent houses, it cannot be for residence, for he would bring in his idolatry, and this is forbidden - "v'Lo Savi So'evah El Beisecha".

21a (Mishnah): In Surya we may rent houses...

Question: Selling is forbidden. Presumably this is a decree lest one sell in Eretz Yisrael. We should likewise forbid renting!

Answer: Renting is forbidden only due to a decree, lest one come to sell. We do not make a decree due to a decree.

(Mishnah): In Chutz la'Aretz we may sell houses and rent fields.

R. Meir decrees only against selling fields, because (in Eretz Yisrael) two Isurim are involved. We do not decree against selling houses.

(Mishnah - R. Yosi): In Eretz Yisrael we may rent houses...

He decrees only against renting fields, because two Isurim are involved with (selling) fields. In Surya he permits selling houses, but not renting. He holds that Kivush Yachid is not considered conquest (Surya is not part of Eretz Yisrael.) He decrees only against selling fields, for two Isurim are involved (in Eretz Yisrael) with fields.

(Mishnah): In Chutz la'Aretz we may sell houses and fields.

Since it is far from Eretz Yisrael, we do not decree at all.

(Rav Yehudah): The Halachah follows R. Yosi.

Rishonim

The Rif (Avodah Zarah 6a,b) brings the Mishnah about renting houses.

Ran (6b DH Masnisin): The Yerushalmi says that where we may sell, one may sell or rent even for residence. If so, where we may only rent, e.g. Surya according to R. Meir or Eretz Yisrael according to R. Yosi, we may not rent for residence. In Chutz la'Aretz one may sell or rent even for residence. This is because "v'Lo Savi So'evah El Beisecha" is an Asmachta. The verse really discusses a Yisrael bringing idolatry into his house. Therefore, Chachamim were more stringent in Eretz Yisrael, in which we are commanded to uproot and eradicate idolatry.

Rambam (Hilchos Avodah Zarah 10:4): Even where one may rent to Nochrim, one may not rent to them for residence, for they will bring in idols, and it says "v'Lo Savi To'evah El Beisecha." One may rent houses to them to make them storehouses.

Ra'avad: One may not rent a house to make it a storehouse of wine. If he did, the wages are forbidden.

Kesef Mishneh: The Rambam did not need to write this. He already taught that all benefit from idolatry is forbidden.

Hagahos Maimoniyos (1): The Ri says that we rent to Nochrim because the Torah forbids only a house in which the Yisrael himself lives. Renting other houses is only mid'Rabanan. Chachamim decreed only in Eretz Yisrael, but not in Chutz la'Aretz. R. Chayim Kohen brings a proof from the Yerushalmi, which says that in a place where the custom is to sell, one may sell even for residence.

Rosh (Avodah Zarah 1:22): Why is the custom to rent to Nochrim, even though they bring idolatry in shortly before death? The Tosefta (2:3) forbids renting houses but permits renting fields, storehouses and inns, even though they bring into all of these! It seems that the Isur to rent is mid'Rabanan. The verse forbids a Yisrael. Chachamim forbade renting to a Nochri, for he will bring idolatry into the Yisrael's house. They forbade only houses, for Nochrim always fixed idols in houses. They permitted fields, storehouses and inns, even though they bring in idolatry, for it was not Kavu'a (fixed). Nowadays, Nochrim bring in idolatry only when one is sick, so it is permitted. Also, even though we hold that rental does not acquire, the Nochrim are in control, and they say that rental is a sale for the time. Even if the owners' house (in which he lives) fell down, he could not evict the tenant.

Tosfos (Avodah Zarah 21a DH Af): The Tosefta connotes that if Nochrim do not bring in idolatry in a Kavu'a way, it is permitted. However, this is not a proof. Perhaps the Torah forbids only idolatry in houses for residence, and there is it is forbidden even Arai (not Kavu'a). R. Chayim Kohen brings a proof from the Yerushalmi, which says that in a place where one may rent, he may not rent for residence, but where the custom is to sell, one may sell or rent for residence. He holds that the Isur is only in "Beisecha", i.e. in Eretz Yisrael. The Ri disagreed. A Yisrael in Chutz la'Aretz is no worse than a Nochri. All agree that a Nochri can acquire land in Chutz la'Aretz! Also regarding Mezuzah it says "Beisecha", and surely this is obligatory in Chutz la'Aretz. One who rents in Chutz la'Aretz is exempt for 30 days. Afterwards, he is obligated, for it looks like his. If it were his, he would be obligated immediately! The Tosefta is unlike the Yerushalmi, for it forbids renting 'here and here', i.e. even where one may sell. The Ri says that the Yerushalmi is a good proof. The Tosefta is not difficult. It forbids renting 'here and here', i.e. Eretz Yisrael and Surya, but in Chutz la'Aretz it is permitted. However, the reasoning is not like R. Chayim Kohen, rather, like R. Elchanan explained. The Torah forbids only a house in which the Yisrael himself lives, even in Eretz Yisrael. This is like Mezuzah, which is obligatory on his house only if he lives in it. The Torah exempts one who rents even in Eretz Yisrael from Mezuzah. Chachamim decreed not to rent to Nochrim lest they bring in idolatry. They decreed only about houses for residence. This is why they permitted renting stables and storehouses. Since renting house in Eretz Yisrael is only mid'Rabanan, Chachamim were lenient in Chutz la'Aretz.

Tosfos R. Elchanan (21a DH Af): Perhaps "v'Lo Savi..." is only mid'Rabanan when the Nochri brings in idolatry against the wishes of the Yisrael. A rental is like a sale according to the opinion that Kinyan Peros is like Kinyan ha'Guf. The decree lest one rent an animal to a Nochri is like the opinion that Kinyan Peros Lav k'Kinyan ha'Guf. This is the Halachah (like Reish Lakish), therefore one must be stringent. 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 me', but one who sold cannot say this.

Poskim

Shulchan Aruch (YD 151:10): Even where one may rent to Nochrim, it is permitted only for the sake of a storehouse or similar uses. One may not rent to them for residence, for they will bring in idols in a fixed way.

Beis Yosef (DH v'Lo): One may sell, for then it is the Nochri's house. Even though rental is in the Nochri's Reshus (jurisdiction), the house is still the Yisrael's

Rema: Nowadays, the custom is to rent even for residence, since Nochrim do not bring idolatry into their homes. One who rents a house from a Nochri should be stringent not to leave the Nochri's idolatry there.

Shach (17): We see that they bring idolatry in! It is difficult to say that we are lenient because it is not real idolatry (merely the custom of their fathers). Perhaps in the days of the Rema they did not bring in idols in a Kavu'a way, like in the days of Rosh and Tur. It seems that we rely on the Rosh's other answer, that according to their law rental is like a sale. Even if the owner's house fell, he cannot evict the tenant. Ra'avan says that since Yisraelim pay taxes from land, the house is not solely the Yisrael's. Tosfos permits in Chutz la'Aretz for this reason.

Note: I did not find such a Tosfos. Tosfos in Kidushin (36b DH Mitzvah) says that land is not considered ours for this reason.

Bach (312 DH Kasav Beis Yosef): Tosfos (Avodah Zarah 15a DH Hashta) says that a rental is a sale for that time only regarding Ona'ah. For other matters, mid'Oraisa it is not a sale. This is why one may not rent his house to a Nochri.