I think that as long as the Pope isn't a Medici (So until 1512.) the Friar is going to survive and his Theocratic Republic too.
Eighteen years of Theocratic Proto-Socialist Republic with CRGCs and popular support would reshape Florence.
A possible consequence is a revision of Renaissance (The Friar didn't burn the works of anybody himself: he convinced the painters who converted and followed him to do it in order to remove those "evil pagan superstitious horrors" and he also encouraged 'em not to stop workin' on religious works.): more and more painters may follow Botticelli and a New Florence's School that refuses nudity and mythological subjects may start.
Also, there were few analogies between the Friar's ideas and Luther's ones.
ITT the Republic would affect history this way IMO:
• 1498 - 1512: New Florence's School blooms led by Botticelli with the help of a young Michelangelo.
• 1508: Michelangelo isn't in Rome and so someone else paints the Sistine Chapel.
• 1512: End of the Republic and 'martyrdom' of Fra' Girolamo Savonarola.
• after 1512: Diaspora of NFS' artists.
• after 1517: Reformation spreads as IOT, but some Protestant groups form in Florence under the influence of the ones in Lucca.
• 1527: Protestants and Catholic dissenters still rebel against the Medicis.
• 1530: Francis Ferrucci still leads the city during the Imperial siege (Michelangelo still helps defending it.).
• after 1530: the city is ""cleaned"" (All Protestants killed.), but the experience is so traumatic nobody in the city would ever forgive the Medicis.
• 1533: after a fire burns the Archbishop's Palace there're few disorders in the "slums" of the city and there're rumors of a rebellion, but nothing happens.
• 1534: a man is hanged for celebrating the death of Pope Clement VII.
• after 1534: history as we know it, but with another artistic movement and its influence on Arts.