No, Shiite can also be used as noun according to Merriam Webster's Dictionary. But Shia is always a noun. The following definitions are from MW's Dictionary.
Shiite (noun): a Muslim of the Shia branch of Islam.
- Shiite adjective
Shia (noun)
1: the Muslims of the branch of Islam comprising sects believing in Ali and the Imams as the only rightful successors of Muhammad and in the concealment and the messianic return of the last recognized Imam.
2: Shiite
3: the branch of Islam formed by the Shia