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Question: 

According to Jewish Law, is it permitted to leave Eretz Yisrael to visit the gravesites of Tzaddikim [the righteous]?

Answer:

Before answering your specific question, let's take a look at the general ruling regarding living in Eretz Yisrael [the Land of Israel], as recorded by the Rambam (Maimonides) in his halachic opus, the "Mishneh Torah."  He states:

Like someone who worships idols? This is a pretty powerful statement. How can it be that leaving the Land of Israel is like serving other gods? In his "Commentary to the Torah," the Ramban (Nachmonides) explains that Hashem (G-d) rules over the Land of Israel by Himself; whereas outside of the Land, He has appointed celestial ministers to rule over the nations. Thus, when a Jew prays or learns Torah in the Land of Israel, his worship and service goes straight up to Hashem. Outside of the Land, however, a Jew's worship goes up to the celestial minister in charge, thus strengthening the foreign nation where he lives.  In this sense, the Jew who lives in the Diaspora is like someone serving other gods (Ramban, Commentary on the Torah, Vayikra, 18:25).

Certainly, one point that this law is making is that the Land of Israel is the Jew's natural, optimum, spiritually healthy place to serve G-d.

The Rambam emphasizes this by stating: "The greatest of Sages would kiss the borders of the Land of Israel, kissing its stones and rolling in its dust, as it says, For her servants desired her stones and cherished her very dust" (Tehillim, 102:15. Rambam, Ibid, 5:10).

Even though it is permitted to leave the Land of Israel to study Torah, to marry, and to make a livelihood, Rabbi Tzvi Yehuda Kook, Rosh Yeshiva of Yeshivat Mercaz HaRav in Jerusalem, would often add that in our time, thank G-d, Israel is the Torah center of the world, there are plenty of religious Israeli girls to marry, and plenty of good jobs in Israel's growing economy, so the need to leave the Land no longer exists.

Regarding the specific question of leaving the Land of Israel to visit the gravesites of holy Sages and Tzaddikim, the"Pri HaAretz," states that even if a person lives in Eretz Yisrael and his soul longs to prostrate himself on the gravesites of Tzaddikim outside of the Land, this is permitted, as long as he plans to return (See Mishna Berurah, Section 568:10, Shaare T'shuva, 20, there). According to the "S'deh Chemed," paying homage to the memory of our holy Sages by visiting their graves is considered an aspect of studying Torah, since our identification with the great rabbis of the past has the power to raise a person in his Torah learning and his worship of G-d. At the same time, there are disagreeing opinions, especially concerning Torah scholars whose study will be hampered by the hardships of traveling, so each case must be judged individually.

Former Chief Rabbi of Israel, Rabbi Mordechai Eliahu, emphasizes that while there is a general understanding

Rebbe Nachman's Tomb