The Gemara in Avoda Zara (13a) states that a cohen is only allowed to leave Israel (because of the gezera of טומאת ארץ העמים) to learn torah, get married, or save his money from a goy. This is brought down l'halacha by the Rif, Rambam, Rosh, Tur and Shulchan Aruch (Siman שס"ט ושע"ב) in Yoreh Deah. However, the Maharshal and in his footsteps the Shach and Taz, say that this din doesn't apply nowadays.
This is very difficult because gezeros apply even if the reason doesn't apply. In fact, the Shvus Yaakov (quoted by the Pischei Teshuva on Shulchan Aruch) disagrees with them and claims that the din applies nowadays. It is clear that the Rif, Rosh, Tur and Shulchan Aruch hold this way as well, otherwise they would have left this din out. The Shvus Yaakov says that a Cohen who lives in Chutz Laaretz is considered an Anus and doesn't have to pick up and move but a Cohen who lives in Israel is only allowed to leave for the reasons listed in the Geara.
R' Akiva Eiger (סימן שס"ט) also is bothered why people don't observe this din. He offers 2 suggestions:
1. A person who has a parnasa in chutz laaretz is no worse then someone going to save his money from a goy
2. Since we are all tamei meis anyway the gezera doesn't apply (I don't understand why).
The Gra in או"ח סימן תקל"א ס"ק ד is also משמע that this din applies nowadays.