In the Igros Moshe, OC 2:113. R’ Moshe answers a question regarding the Har HaBayis.
The question was raised about why the Shulchan Aruch and the Tur both don’t mention an issur of spitting on the Har HaBayis in modern times. The person who asked gave a possible explanation that the question is irrelevant since nobody is allowed on the Har HaBayis in modern times. R’ Moshe rejects that answer, and says that there are parts where people are allowed to walk. He then gives a different answer regarding the spitting question.
From this, it appears that R’ Moshe held it was allowed to go on the Har HaBayis – indeed, R’ Tendler zt”l said many times that R’ Moshe was well aware of R’ Tendler’s going onto the Har HaBayis, and never attempted to dissuade him from doing so.