Gemara

(Beraisa): One who stays in an inn in Eretz Yisrael, or rents a house in Chutz la'Aretz is exempt from Mezuzah for 30 days. After this, he is obligated.

One who rents a house in Eretz Yisrael is obligated immediately, for the sake of settling Eretz Yisrael.

Bava Metzia 101b (Rav Mesharshiya): The one who lives in a house is obligated to affix a Mezuzah. The Mitzvah is Chovas ha'Dar (incumbent on the resident).

Shabbos 131a (Beraisa): All agree that if on Shabbos one put Tzitzis on his garment or fixed a Mezuzah in his doorway, he is Chayav.

131b (Rav Nachman): This is because he could make his garment or house Hefker (therefore, the Mitzvah is not obligatory, and it does not override Shabbos).

Chulin 135b (Rava): One might have thought that partners (in a house) need not affix a Mezuzah, because it says "Beisecha" - not of partners "Lema'an Yirbu Yemeichem" teaches that partners are liable;

"Beisecha" teaches that a Mezuzah is placed on the right doorpost (We read this "Bi'asecha", the way one enters a house, with his right foot first.)

Rishonim

Rif and Rosh (Hilchos Mezuzah 6b and Siman 17): One who rents a house in Chutz la'Aretz, or lives in an inn even in Eretz Yisrael, is exempt from Mezuzah for 30 days. One who rents in Eretz Yisrael is obligated immediately, due to settlement of Eretz Yisrael.

Rambam (Hilchos Mezuzah 5:10): One who rents a house in Chutz la'Aretz, or lives in an inn even in Eretz Yisrael, is exempt from Mezuzah for 30 days. One who rents a house in Eretz Yisrael is obligated immediately.

Tosfos (Bava Metzia 101b DH Lo): Mazikin (Shedim, i.e. damagers with similarities to humans and angels) enter a house without a Mezuzah.

Mordechai (Avodah Zarah 810): Mezuzah is Chovas ha'Dar. One is liable only if he owns it and lives there. Do not say that a resident is liable even if he does not own it, for in Menachos we exempt one who borrows or rents a house up to 30 days. After this, he is liable. If it were mid'Oraisa, even if it is not his house, he would be liable immediately! Rather, "Beisecha" excludes another's house. One is liable after 30 days, for people think that it is his. Alternatively, the Torah obligates even in another's house. One is exempt for 30 days, for until then he is not fixed there.

Tosfos (44a DH Talis): Regarding Mezuzah, it is unreasonable to say that Beisecha excludes another's house, for it is for Shemirah. Rather, until 30 days it is not considered his house. One is considered a resident of a city after living there 30 days (Bava Basra 8a). Perhaps Mezuzah after 30 days is mid'Rabanan. A second verse says Beisecha. When we say that it is Chovas ha'Dar, this is not to obligate a renter, rather, to exempt the owner.

Mordechai (Menachos 962, citing R. Meir bar Baruch): I am assured that no Mazik can enter any house with a Kosher Mezuzah.

Poskim

Shulchan Aruch (YD 286:22): One who rents a house in Chutz la'Aretz, or lives in an inn even in Eretz Yisrael, is exempt from Mezuzah for 30 days. One who rents in Eretz Yisrael is obligated immediately, due to settlement of Eretz Yisrael.

Beis Yosef (DH ha'Socher): Rashi explains that we obligate a renter in Eretz Yisrael immediately, to promote settling Eretz Yisrael. Once he affixed a Mezuzah he may not take it, even if he leaves. Therefore, he is reluctant to leave, due to the toil of another Mezuzah. Even if he leaves, another will be quick to rent it, since it has a Mezuzah.

Shach (28): Within 30 days he is exempt because it is not yet called a residence.

R. Akiva Eiger (Teshuvah 66): A renter is obligated in Mezuzah. This is not because rental acquires and it is called his house. It acquires only Peros, but not the land itself! Tosfos leans to say that even after 30 days, the Chiyuv is only mid'Rabanan, for it looks like his house. Tosfos Avodah Zarah (21a DH Ha) and the Rosh hold like this. Why does the Shach explain that within 30 days it is not called a residence? Perhaps he explains the Mechaber, who obligates a house owned together with a Nochri. He must hold that one need not own the house, and the Torah obligates a renter after 30 days. This is difficult. If the Beis Yosef held that partnership with a Nochri depends on whether the Torah obligates a renter, he would not have obligated (like the Rashba) against the Mordechai, Rosh and Tosfos. Really, he agrees that the Torah exempts a renter. We hold that rental does not acquire. One is liable after 30 days only because it looks like his. If so, surely if a Nochri pays rent to live with a Yisrael, this does not exempt from Mezuzah.

Nachalas Tzvi: It seems that even if he rented for a year, he is exempt until 30 days. Also the Beraisa connotes like this. The Sidur of the Nesivos ha'Mishpat says 'one who rents a dwelling up to 30 days is exempt.' This implies that if he rents for longer, he is liable immediately. What is his source? We hold that after 30 days he is obligated mid'Rabanan because it looks like his. Until then, he is exempt! Perhaps he learns from the Shach, that after 30 days it is called a residence. Regarding Tzedakah, one is liable after living in the city 30 days, or immediately if he intends to settle there (Bava Basra 8a). However, R. Akiva Eiger showed that this is unlike the conclusion of Tosfos and the Rosh.

Question (Hilchos Ketanos 95): If one wants to put up a Mezuzah and bless before 30 days, is this permitted?

Answer (Beis Yehudah 19): Some say that women may bless on a Mitzvas Aseh sheha'Zman Gerama (it applies only at certain times). Even though they are exempt, since they receive reward they may bless. The same applies here. However, perhaps even the opinion that forbids them to bless could permit here. Women will never be obligated in these Mitzvos, but a tenant will become obligated.For the same reason, this is unlike the argument about whether a Beis Medrash requires a Mezuzah (Sa'if 10). It is unlike putting Tzitzis on a borrowed garment, for one may not bless on another's garment. When one rents a house, it is his from now. This is like a minor, who may bless now, for later he will be obligated. However, one should not rely on this in practice.

Nachalas Tzvi: Eshel Avraham (brought in Otzar Meforshim Shulchan Aruch ha'Shalem on OC 14:2) says that surely, one may put up a Mezuzah and bless before 30 days, just like a borrowed Talis is exempt until 30 days, and the Magen Avraham (14:5) brings from Tosfos and the Rosh that one who put on Tzitzis and blesses does not lose. I say that it is better to wait until 30 days. Perhaps Tzitzis is different, for one blesses on it every day, and he will bless also after it is liable. The Mizrachi (Bereishis 17:25) says that Avraham did not do Bris Milah until he was commanded, for it is a greater Mitzvah when he is commanded. Zichron Yosef (YD Sof Siman 26) says that a minor does all Mitzvos from the age of Chinuch, except for Mitzvos (e.g. Pidyon ha'Ben) that he can do only once. If one rented a house, put up a Mezuzah, and later bought the house, perhaps he should bless again, for now he is obligated mid'Oraisa. This is like R. Akiva Eiger (brought in Pischei Teshuvah 291:4); the Birkei Yosef exempts.

Birkei Yosef: If one was in a house 29 days, left, and returned, do the first days join to obligate him the very day he returns? Beis David (146) was unsure about this.

Pischei Teshuvah (18): I say that this is just like a Talis. The Nimukei Yosef (Hilchos Tzitzis, brought in Rema 14:3) says that if one borrowed a Talis, returned it before 30 days, and borrowed it again, it does not join. We need 30 consecutive days.

Rema: One who borrows a house is like a renter.

Beis Yosef (ibid.): R. Mano'ach says that one who borrows a house in Chutz la'Aretz is like a renter. The Gemara connotes like this.

Gra (19): In Shabbos we say that one could make a Talis or house Hefker, i.e. one is exempt even if he (borrows and) wears it or lives in it. The Gemara could have said that "Beiseicha" excludes one who lives in another's house. Rather, it taught another law (putting it on the right side).