Ciao Pauli,
I take it you've already seen the map with the parking lots in Florence found here:
http://www.visitflorence.com/florence-maps/parking-lots-in-florence.html
While both the InCoop parking and the one at Piazzale Michelangelo are free, the first is for a private supermarket and the gates get closed when the super market closes so that is out. I think it might get towed if left overnight. The one at the Piazzale on the other hand has limited time on the parking spaces.... I think a car staying overnight would be noticed, fined or towed.... (but I will have to find out about this for sure). A safer bet would be to use one of the parking lots listed on that page above or go with a private garage near your hotel, especially if you're planning to use the car during those 5 days.
I have another suggestion: if you're planning to arrive in Pisa and right away coming to Florence for 4-5 days, don't get the car rental in Pisa - catch the train from the Pisa airport into Florence's SMN train station. Your hotel is less than 15min from the train station. You'll just be paying daily for a car you don't need at all in Florence... Right in the same area where you're staying in Florence are many of the car rental agencies, you can rent a car from there once you need it.
If you're planning to visit other part of Tuscany, try to do them before Florence or after it, so that you only have it when you really need it.