It depends on how long you are going for. If you are going for less than a month then you do need to put up a Mezuzah. If you are there for more than a month then you need to.
Explanation:
The Gemara in Menachot 44a: One who lives in an inn in Eretz Yisrael and one who rents in Chutz LaAretz for the first thirty days he is exempt from putting up Mezuzah and from there on he is obligated but one who rents in Eretz Yisrael puts up a Mezuzah immediately because of ‘Settling Eretz Yisrael.”
The Gemara says one who rents for less than a month in Chutz LaAretz is exempt from putting up a Mezuzah.
The commentaries argue what the difference is in Chutz LaAretz between thirty days or less.
Tosfot (dh Tallit) in the first answer says that even though it says by a Mezuzah “Uketavtem al Mezuzot beitecha ” (Write on the doorpost of your house) it is saying that on the doorposts of your own house are you obligated since a Mezuzah is an obligation on the person living there not on the house, the fore it does to make sense to say only the house you won which you live in since it depends if a person is living in a place which provides him shelter. “Your house’ only comes to tell you that you attach the Mezuzah on the right side “Biatecha” the ways you enter is what the Gemara itself says on daf 34a. before thirty days you are exempt since it is not a living space if he did not live there thirty days. As the Gemara in Baba Batra 8a says one is not considered living in the city (for municipal obligations) until thirty days. After thirty days you are obligated to hang a Mezuzah from the Torah.
The Minchat Chinuch sees the same in the Sefer Hachinuch (Mitzvah 423) who writes that if a renter doesn’t hang a Mezuzah after thirty days he nullifies a mitzvah
Tosfot has a second answer that a house which is not yours is excluded from the passuk of Beitecha, just like we understand by Tzitzit from the word Kesutacha, that it has to be your clothing. The obligation in Chutz LaAretz for renting after thirty days is only Rabbinic.
The Chamudei Daniel (Menachot) explains the difference between the two answers of Tosfot is in a case where one rented a house where it is unclear where it needs a Mezuzah, like a shed. According to the first answer of Tosfot that the obligation of Mezuzah by aren’t from the Torah then he must put the Mezuzah anyway after thirty days. However according to the second answer the obligation is only rabbinic, therefore in the case of doubt he will not have to put a mezuzah.
It is clear though that according to both answers of Tosfot the obligation of hanging a Mezuzah in Chutz LaAretz is only after thirty days. The Meiri though (Shabbat 148) quotes those who say the obligation to put a mezuzah after thirty days is only when he entered the apartment with the intention to stay for less than thirty and then he stayed on. However if he intends to live there, he is obligated immediately.
To recap there are three opinions;
Tosfot first answer-A renter for less than thirty days is exempt from Mezuzah but obligated from the Rosh afterward.
Tosfot second answer, Rosh, and Rashba-A renter in Chutz LaAretz is never obligated from the Torah but after thirty days he is obligated from Rabbis to put up a Mezuzah.
The Meiri that under thirty is exempt if he intended to stay for more than thirty days then he is obligated after thirty. If he intends to live there he is obligated immediately.
It seems according to everyone there is no obligation to put up Mezuzah in Chutz LaAretz if your renting. So rules the Shulchan Aruch (Yoreh Deah 286:22)
If one feels more comfortable with putting up Mezuzah less then thirty day, can he say a bracha?
The Shulchan Aruch Orach Chaim Siman 14 by borrowed Tzitzit writers:
The Magen Avraham comments If one put son Tzitzit Tosfot and the Rosh says if he says a bracha it is not in vain, like women who can say brachot on time bound mitzvot corm which they are exempt.
The Magen Avraham says that even though he borrowed the Tallit less then thirty day he is exempt from tzitzit but he may say a bracha. The Levushei Serad explains that one does not say bracha then exempt only when everyone else is exempted. So when men are obligated and women not, women may make a bracha anyway. So too by a borrowed Tallit even though the borrower is exempt from putting tzitzit since the owner is obligated the borrower may also make a bracha.
The Eshel Avraham writers the one puts up a Mezuzah in Chutz LaAretz es than thirty days may make a bracha.
However one may ask that in the case of the Mezuzah everyone is exempt from the mitzvah when he rents for less than thirty days, who can make a bracha? Maybe since in Eretz Yisrael one is obligated less than thirty days he may say a bracha. However the Shulchan Aruch in Orach Chaim Siman 17 writes that women are exempt from timebound mitzvot and only the Rema there mentions that they may Shulchan Aruch bracha. Therefore the custom of the Sephardim is that women do not say a bracha. So too here also well It may be only Ashkenazim may say bracha.
