Joseph, look at Igros Moshe, OC 2:113. R’ Moshe answers a questions 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 has 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.
There is a video of R’ Tendler on the Har HaBayis, where he comments on a mitzvah he tries to do there – there is a tree growing there, and he breaks off a branch and takes it with him. He says no trees are supposed to be grown on the Har HaBayis, and he therefore tries to minimize the growth there. He said he takes the branch home, and burns it.