The Iranian Monarchy was overthrown following the Islamic Revolution in 1979 with the Shah Mohammad Reza Pahlavi going into exile.
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Line of succession to the Pahlavi claim
Under the Pahlavi Dynasty the law of succession stated the Shah must profess the Islamic faith, his mother must be an Iranian citizen, a Muslim and not descended from the previous Qajar dynasty which rules out the sons of Reza Shah by his fifth and sixth wives and their male line descendants.
Given that the modern claim calls for a new constitution, and thus the bylaws need not necessarily be in accordance with the previous constitution in the event the monarchy is restored new succession rules may be established.
Line of succession to the Qajar claim
The Qajar dynasty was deposed in 1925 with Reza Shah ascending the Sun Throne. The Iranian Constitution of 1906 set out the succession for princes whose mother is a Qajar princess of Persian descent. Only males are allowed to succeed.