by Reb Akiva at Mystical Paths
A significant halachic MYTH has developed over the last 40 years, that it is ASSUR (prohibited) for a religious Jew to ascent Har HaBayit - the Temple Mount, former site of the holy Beis HaMikdash, the Holy Temple of G-d, and current site of a number of mosques including the Dome of the Rock and Al Aqsa (not the same thing btw).
What has happened in the last 40 years, since Har HaBayit entered Israeli control (more or less) is this: the government discourages people from going up due to religious-political conflict reasons, some authorities are concerned about going up due to the possible errors of entering forbidden areas (where the Holy of Holies once stood - and it's exact former position is not known) and the severe spiritual results of doing so, and some authorities avoid the topic as not appropriate to discuss or consider prior to the coming of Moshiach. Reb Gutman Locks made the further point to me that as a site becomes familiar it becomes somewhat a tourist attraction and not treated with appropriate holiness - and for Har HaBayit this would truly be a terrible thing.
That said, as we clearly see in the video below, it is NOT prohibited. In the video below, HaRav Moshe Dovid Tendler, shlita, posek (halachic decisor) and son-in-law of HaRav Moshe Feinstein, zt"l, tours Har HaBayit, speaks of it's significance, what's permitted and what is not. It's a bit long (at 17 minutes) but WELL worth taking the time to view. Credit to Israel National News, who's Yishai Fleisher is seen interviewing the rav, and the Temple Institute, who conducts the Jewish KOSHER visits to the holiest site in Judaism.
In the beginning of the video, the rav speaks briefly about the Israeli police instructions, which specific FORBID a JEW from saying a tefilah - prayer or acting in any prayerful way on Har HaBayit, due to the concern it may offend the Muslim presence.
BS"D - בסיעתא דשמיא