Living in Israel - Rav Tzvi Yehuda HaKohen Kook

In this brief teshuva, cited in Techumin volume 3, Rav Tzvi Yehuda affirms that there is definitely a mitzva to live in Israel, as per the view of Ramban. He then goes further and notes that just like one is required to give his life for any mitzva in a situation where he is being forced not to do it [my note: specifically in times of religious persecution], even more so in the case of living in Israel, which is a mitzva that is seen as tantamount to all other mitzvot, one must make whatever sacrifices necessary to fulfill it. This mitzva even goes beyond other mitzvot in this regard, in that a person must put himself into a potentially dangerous situation of living in Israel.

Living in Israel - Igrot Moshe Even HaEzer 1:102

In the last paragraph of a teshuva about a different topic (written in 1952), Rav Moshe Feinstein addresses the issue of whether or not there is a mitzva to live in Israel nowadays. He says that most poskim feel that it is a mitzva. However, he notes that these days it is not a mitzva chiyuvit (one that a person must strive to fulfill in all cases), but rather it is a mitzva kiyumit (a mitzva that one gets credit for doing if he does it, but does not have to put himself in a situation where it becomes obligatory). In making this ruling, Rav Moshe relies on the view of a Ba'al Tosafot named Rav Chaim cited in Tosafot Ketubot 110b, who notes that nowadays (meaning already 800 years ago) there was not a requirement to live in Israel since doing so would potentially trigger the requirement to keep a variety of mitzvot connected to the land, and since we are not fully sure how to keep those mitzvot or if we even can, thus it might be better to avoid such a situation.