From an Orthodox perspective, the question isn't so much whether we have to visit Israel, it's whether or why we can find justification for not living in Israel. Famously, Maimonides did not include an obligation to live in Israel in his list of mitsvot, a lacuna that Nachmanides objected to, and did include in his own list. Several explanations for Maimonides' view have been offered; I find most convincing the one that notes that Maimonides did not include in his list those commandments that overarched several others-- since the commandment to live in Israel underlies many others, he would have seen it as such a mitsvah kollelet, a general mitsvah. I say this because he does, in his Code, include many of the same Talmudic statements about the value of living in Israel that Ramban had used to support his claim that it was a mitsvah.
Of course, whichever way we come down on that question, the reality is that many Jews have lived outside Israel throughout history (including Maimonides and Nachmanides for the vast majority of their lives). Granting that there are valid reasons for this (I myself have not yet made it to living in Israel), we can ponder the question of visiting. There is an opinion that says that just walking in the Land of Israel constitutes a fulfillment of the obligation to settle the Land, so that visiting would be a mitsvah in that sense. In addition, while the obligation of aliyah le-regel, going to Jerusalem for the three major holidays, is fairly clearly connected to the existence of a Temple, Maimonides' view that the Divine Presence never left Jerusalem would suggest a value to being in Jerusalem for these holidays even today.
There is also an element of supporting other Jews' valuable endeavors by visiting Israel. However we excuse remaining outside the Land, our brethren who are building the Land and the State are unequivocally fulfilling a high ideal and goal of God's world, and it behooves us to support this and all other worthy endeavors to the greatest extent of our abilities. Visiting-- especially when it is difficult for others to do so, but at all times as well-- is one fairly minimal way to give this support (and has a value far beyond just writing a check, valuable as that is).
Finally, I would note that the Sefer HaHinuch, a compilation of the Torah's commandments, assumed that several agricultural obligations, such as eating a secondary tithe in Jerusalem, were actually excuses to force us to go to Jerusalem, since the assumption was that Torah would emanate from there. Expanding that, it is certainly arguable that, for almost all of us, the experience of Torah and of God running the world is heightened, certainly in Jerusalem, but to some extent throughout the Land. Visiting (at least once, if not annually or more) puts us in touch with an experience of the world, of truths about the world, that we rarely access elsewhere, a truth that has been noted by many Jews who have visited Israel, indepedent of any prior religious feeling.
So: I'm not sure if, in technical terms, I can show you an obligation to visit Israel once in your life. I can show you reason to believe that Jews are supposed to be living in Israel, reason to believe that, in the absence of that, there is a value to experiencing Israel on the holidays and in general. If that's enough to get you to visit Israel regularly, all the better.
It is unquestionably an obligation for Jews to support the continued stregnthening of the Jewish community in our ancestral homeland. One excellent possible component of that support, if not the only one, would be a visit to Eretz Israel, supporting your brothers and sisters with money and love. Moreover, visiting the Land of Israel -- seeing the places where our ancestors and sages lived and worked and built and celebrated and fought and sacrificed -- is all but certain to strengthen one's own Jewish identity today.
That said, there is probably no formal obligation in Jewish tradition to visit the Land of Israel, at least if one is not intending to settle there (at least semi-) permanently.
Medieval and ancient sages debated whether there was a formal commandment for every Jew to live in the homeland. They all agreed it was good to live there. But was it required? Even for those who thought it was itself a mitzvah to live in Israel, it is unlikely that they would have regarded it as an obligation to visit there in the merely temporary pilgrimage of a visit.
But even if it is not a mitzvah to visit, it is still a blessing and a privilege to be able to do so. Those who can afford such a trip should take every opportunity they can to visit. As the Talmud [Ketubot 110b] says: Anyone privileged to walk four paces in the Land of Israel is certainly worthy of the world to come. Amen to that.
There is no specific mitzvah to visit Israel in any of the halachic works that I have ever read. Indeed, after the destruction of the Temple by the Babylonians and the subsequent rebuilding as described in the books of Ezra and Nehemiah, there were plenty of Jews who stayed in Babylonian exile and did not come back to Eretz Yisrael. If there was a halachic mandate to move back, more probably would have.
Unlike the Muslims who have pillar of Islam being the Hajj – the pilgrimage to Mecca – Jews do not have such a thing. I have sometimes wondered why this is, especially since Jerusalem plays such an important part of the Jewish scriptures. Perhaps it was a fear of deifying the city and turning it into an idol of sorts.
But, having said that, there are many stories of Jews throughout history who are drawn to the Land as it became part of their neshama – their very soul. Judah HaLevi, the great mediaeval poet and philosopher pined away for Jerusalem with the words, ‘my heart is in the East.’ In recent years we have seen Ethiopians, Russians, Yemeni and Iraqi Jews find safety and security in the Land. But getting there was for religious reasons which are different than halachic reasons.
Of course, from the US getting to Israel means going to the airport, falling asleep on the plane and waking up in Tel Aviv. So even though it is not a religious obligation to daven at the Kotel, to shop on Dizengoff, to wake up in Sfat, to hike the Golan, to swim in the Dead Sea, to sail in the Kinneret, and to visit friends in Ma’ale Adumim, with it being so easy, the real question is ‘what is taking you so long?’ My rabbinic advice: go and take your family and, in Israel, you may find even more of your Jewish neshama.