The late-Shah accorded Israel an unofficial embassy status in Tehran and the Israeli representative all the privileges, diplomatic status and respect of an Ambassador.
Having been there on several occasions (so long ago), I remember the video security camera which was rare in those days and an intercom connection to their security before they let you in and if I remember correctly a narrow metal (steel?) door in the outer wall of a modest enough building on a narrowish street.
Eventually taken over by Yasser Arafart as his Tehran office after the Khomeini revolution.
The Israeli “embassy” in Tehran provided visas for people tavelling to Israel. No objections by the Shah other than not “officially” recognizing the Israeli Mission as an Embassy.
It was provided on a separate piece of paper and not stamped into Iranian passports. Not a demand by the late-Shah but by those wanting a visa and later travelling on the same passport to other countries including Aab ones, who would refuse entry if an Israeli visa was in it.
The probable link to the Communist comment is that Soviet Jews/Emigrants were often connected to the banned Communist Tudeh Party and a free flow of potential Soviet spies was made as difficult as possible for security reasons.
Israel’s emigrants/agents from the Soviet Union inside Israel made quarantining Soviet plants meeting Iranians in Israel very difficult, so some obstacles were in place.