Reza Pahlavi is the "Crown Prince of Iran" ...

... in, at most, the same sense that Franz von Bayern is "by the Grace of God, of the United Kingdom of Great Britain and Northern Ireland and of His other Realms and Territories King, Head of the Commonwealth, Defender of the Faith."

But really, not even that much.

The form of the regime that James Francis Edward Stuart and his Jacobite "pretenders" attempted to assume chiefdom of  by "succession," starting in 1701 with the death of the exiled James II/VII, remains in existence.

The Iranian monarchy itself -- not just a particular Shah -- was extirpated nearly half a century ago.

Pahlavi's claim is pretty much a dead parrot.