Gemara

A Yavam asked whether he should go to Chutz la'Aretz to do Yibum.

R. Chanina: His brother died for marrying a Nochris (i.e. a resident of Chutz la'Aretz). Blessed be Hash-m, Who killed him for that. Will the Yavam do the same?!

Mo'ed Katan 13b (Mishnah): One who came from overseas on Chol ha'Mo'ed may shave.

14a (Beraisa - R. Yehudah): He may not shave, for he had no Heter to leave.

(Rava): If he left to travel, all forbid. If he left (to earn money necessary) for food, all permit. They argue about if he left to earn extra money.

Avodah Zarah 13a (Beraisa): A Kohen may go to Chutz la'Aretz, even though this makes him Tamei mid'Rabanan, to plead his case. He may become Tamei (mid'Rabanan) to learn Torah or marry a woman.

R. Yehudah says, if he can find a Rebbi here, he may not become Tamei to learn elsewhere.

R. Yosi permits this, for perhaps he will learn better from the Rebbi there;

Yosef ha'Kohen followed his Rebbi to Tzidon (in Chutz la'Aretz) to learn.

(R. Yochanan): The Halachah follows R. Yosi.

Nazir 65b - Question: If a third Mes was found after two Mesim were found and relocated, the area should be established to be a burial site!

Answer (Reish Lakish): They found an excuse to establish Eretz Yisrael to be Tahor.

Rishonim

Rif and Rosh (Avodah Zarah 3b and 1:12, also Sof Hilchos Tum'ah): A Kohen may became Tamei mid'Rabanan by going to Chutz la'Aretz to plead his case, to learn Torah or marry a woman. The Halachah follows R. Yosi who permits even if he can find a Rebbi here, for perhaps he will learn better from the Rebbi there.

Rambam (Hilchos Evel 3:13): If a Kohen became Tamei through a Beis ha'Peras or Chutz la'Aretz, he is lashed mid'Rabanan.

Rosh (Kesuvos 2:5): The Yerushalmi (Kil'ayim 40b) says that we are not particular about Mesim or Kil'ayim in the Beis Medrash. I.e., a Kohen may remain in Ohel ha'Mes for the sake of Talmud Torah. The Bavli disagrees and permits only Beis ha'Peras and Chutz la'Aretz, which are Tum'os mid'Rabanan.

Rosh (Mo'ed Katan 3:1): The Ra'avad says that Chachamim and R. Yehudah argue about one who went to Chutz la'Aretz. The Yerushalmi supports him. It says that R. Yehudah forbids going in the Mediterranean Sea. Therefore, one who went to Chutz la'Aretz is forbidden, since Chachamim did not permit him to leave. This is like the Kohen who asked to go to Chutz la'Aretz to do Yibum, and R. Chanina forbade him. Presumably, this is correct. The Gemara suggested that R. Yehudah forbids gong for food. Surely, one may go on the sea for food. All merchants do so! Rather, they argue about going to Chutz la'Aretz.

Rebuttal (Gra OC 531:4 DH v'Hu): The Yerushalmi is unlike the Ra'avad! It connotes that R. Yehudah discusses one who came from the sea, not from Chutz la'Aretz. It is forbidden due to danger. Only a Kohen may not go to Chutz la'Aretz. This is unlike the Bavli, which forbids even Yisraelim.

Defense (Rashash): The text of the Yerushalmi (10a) says 'therefore, a Kohen who went to Chutz la'Aretz is forbidden...' I.e., even Chachamim agree (Korban ha'Edah).

Note: The Gra had the same text, but he explains like the Pnei Moshe, that therefore, R. Yehudah forbids a Kohen who went to Chutz la'Aretz.

Poskim

Shulchan Aruch (YD 369:1): A Kohen may not become Tamei by entering Chutz la'Aretz.

Prishah (3, citing Maharshal), Bach, Shach (2), Taz (4), Chachmas Adam (159:1): This refers to when Eretz Yisrael was Tahor. Then it was forbidden to go to Chutz la'Aretz, where Nochrim bury their dead everywhere.

Rebuttal (Shevus Yakov 1:85): The Rif, Rosh, and Shulchan Aruch would not bring a law that will not apply until Moshi'ach! It seems that Kohanim live in Chutz la'Aretz for the sake of income, just like one may go to Chutz la'Aretz to recover his money. Not everyone can manage in Eretz Yisrael, and there are dangers. However, a Kohen in Eretz Yisrael should leave only for needs of income, learning and to get married. Tosfos says that the decree on Chutz la'Aretz was to stop Kohanim from leaving Eretz Yisrael! Kohanim in Chutz la'Aretz are already Tamei Mes mid'Rabanan, so they need not be careful of mid'Rabanan stringencies of Tum'as Ohel. This is why a Tosefta (cited in Tosfos Yevamos 61a DH mi'Maga) permits Kohanim in Chutz la'Aretz to enter houses of Nochrim.

R. Akiva Eiger (DH v'Chol): Nowadays Kohanim are not careful about this. Perhaps they are lenient for the needs of livelihood. Alternatively, we are lenient because we are all Tamei nowadays.

Chasam Sofer (YD 337 DH Lomar and v'Hinei): Tosfos (Nazir 54b DH Eretz) gives different reasons for the decree of Tum'ah on Chutz la'Aretz. If it was due to Mesei Mabul (those who died in the flood), or the many Yisraelim killed in Chutz la'Aretz, Eretz Yisrael was different because they cleared away the Mesim for the sake of Taharos. This was after the first Churban, but not after the second, so nowadays Eretz Yisrael and Chutz la'Aretz are the same. However, according to the other reasons for the decree, even today Kohanim may not go to Chutz la'Aretz. Reish Lakish taught that 'they found an excuse to be Metaher Eretz Yisrael. This applies also to Chutz la'Aretz. If not, we should be concerned for bones from Mesei Mabul in Chutz la'Aretz (Zevachim 113a)! Rather, he mentioned Eretz Yisrael because Chachamim decreed Tum'ah on Chutz la'Aretz, and Kohanim could not go there in any case. Nowadays, Kohanim in Chutz la'Aretz need not be concerned for Tum'os mid'Rabanan, especially according to the opinion that the decree on Chutz la'Aretz was due to Mesei Mabul or Yisraelim killed there.

Shulchan Aruch (372:1): Even though a Kohen may not go to Chutz la'Aretz, if he needs to go to get married or learn Torah, and there is no alternative path, he may go through it. This is even if he could learn elsewhere

Beis Yosef (DH v'Chosvu): Tosfos (Avodah Zarah 13a DH Lilmod) says that the Heter to go to Chutz la'Aretz for marriage or learning is only if he intends to return to Eretz Yisrael, like we find in Kesuvos (111a). Tosfos says that the Heter is only for these two great Mitzvos. Megilah 27a supports this. (R. Yochanan permits selling a Sefer Torah only to learn Torah or to marry a woman. This is because learning brings to fulfilling Mitzvos, and "Lasheves Yetzarah".) However, the She'altos (Emor 103 DH Berem) permits even these light Mitzvos, and all the more so other Mitzvos.

She'alas Shalom: Surely, the She'altos agrees that these are the most important Mitzvos! We permit these even if he could fulfill them without becoming Tamei, e.g. he can learn better from a Rebbi in Chutz la'Aretz.

Divrei Chamudos (on Rosh Hilchos Tum'ah, 37): The Rif and Rosh seem to hold like the She'altos.