Those are much broader questions, then. Just looking at your word "Now", seems to imply that the access is variable, historically, depending on who is politically/militarily in charge.

I think I remember some Jewish restrictions on who may enter temple, especially inner-sanctums. I don't recall specific Christian prohibitions against non-believers being present in churches. And I confess woeful ignorance of Qu'ranic injunctions... so I guess I'll sit this one out as a reader, with the parting thought that it seems to be political, not religeous - and all tied up in the Jerusalem/East Jerusalem - capital of Israel/capital of Palestine imbroglio.