Is a Kohain allowed to enter the Mearas HaMachpela?
In Iturei Cohanim (Sivan 5766 #261), Rav Aviner was asked, is it permissible for a cohain to enter Ma’arat Ha-Machpelah? He answered that there is a dispute, but Maran Ha-Rav Kook did not enter. Similarly, in the weekly parashah sheet "Olam Katan" (#169), Rav Aviner was asked, is it permissible for a cohain to enter Ma’arat Ha-Machpelah to recite Selichot? He answered that there is a dispute, but Maran Ha-Rav Kook did not enter. In Shut She’eilat Shlomo (vol. 3 #329), Rav Aviner also writes that although there are authorities who permit cohanim to enter "Kivrei Tzaddikim," since the righteous are called "living even in their death," the accepted halachah is that it is forbidden. There are also authorities who allow cohanim to visit Maarat Ha-Machpelah and Kever Rachel, because they were built in a way that the cohanim would not become impure; but the acceptable halachah for this is also that it is forbidden. Therefore, we say that cohanim should not enter "Kivrei Tzaddikim," but we can defend the practice of those who act in this way, especially entering Maarat Ha-Machpelah and Kever Rachel.
Rav Ovadia (Yechaveh Daat 4:58) was opposed, but Rav Mordechai Eliyahu and Rav Dov Lior argued the structure obstructs any Tuma from going up. See Rav Dov Lior's Devar Chevron Yoreh Deah 210, Taharat Sdeh HaMachpelah, Rav Shlomo Dichovsky in HaMaayan 235 (Tishri 5781), Otzar Forums, and this shiur for more details.