According to the Wikipedia article about the Shahnameh, the Persian poet Ferdowsi wrote this epic "for Sultan Mahmud of Ghazni between c. 977 and 1010 CE". This was long before the introduction of the printing press in Persia or Iran. According to the Wikipedia article Global spread of the printing press, the first book printed in Iran was "Սաղմոս ի Դավիթ (Saghmos i Davit - Psalter) in Armenian", in 1636. The other books listed there were printed in the early 19th century. There are a number of magnificent manuscripts, such as the Shahnameh of Rashida and the Windsor Shahnameh, both dating from first half of the 17th century, the same period when the first printed book was produced in Iran.
I have not found evidence that printing was at some point prohibited by any of the dynasties that ruled Iran between 1636 and the early 20th century, unlike the Ottoman Empire, where printing in Arabic script was prohibited in 1483 (see my question When was Mem and Zin / Mam û Zîn first printed?).
Work on the first scholarly edition began in 1808, but the first eight volumes of this edition were published in Kolkata, i.e. outside Persia. The first complete scholarly edition, published in 1829, "was based on a comparison of 17 manuscript copies", i.e. not on existing printed copies. Does this mean that no printed versions of the Shahnameh were produced before the first scholarly edition? Or were early 19th-century scholars unaware of printed editions that existed?