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.