Clean is an adjective, modifying he.
O'Connor - or, more accurately, her character - is treating burn as analogous to wash: the evils which come of the Lord are experienced by him as a fire which does not consume but cleanses, burns off what is impure and leaves him clean. So clean is an "object complement" of the verb in burn the prophet clean and a "subject complement" in he himself had been burned clean and burned clean again.
The notion of cleansing fire runs throughout the Hebrew Bible, and is embraced by the Christian. For instance, John the Baptist is reported to have said:
I indeed baptize you with water unto repentance: but He that cometh after me is mightier than I, whose shoes I am not worthy to bear: he shall baptize you with the Holy Ghost, and with fire. -Matthew 3:11