Renting a car in Florence and returning it in Rome

First of all make sure you have a valid IDP. They are valid only for one year.
Where are you staying the night before your flight? If you are staying at the airport Hilton, you should return it at the airport the day before your flight when you arrive. Parking at the Hilton is not free and you can easily walk from the rental return to the hotel.
If you are staying downtown, you should return the car also when you arrive at a city rental office (for example at termini or a Tiburtina station, or any location convenient for your hotel). Don’t keep the car overnight in the city center, because the parking fees and rental charges for the extra day will easily equate the taxi ride to the airport on the morning of departure (€48 flat fare). Also large portions of the city center are off limits to non residents’ cars.
If you are staying at an out of town hotel with free parking, somewhere in the outskirts of Rome, then you might consider returning the car at the airport on the morning of your flight. Give yourself an extra 20-30 min for the car return paperwork and the walk to the terminal.

Or return car to airport and then go into Rome for the night if you don’t have an early flight. If early flight, we always stay at the Hilton, walkable from car rental return and adjacent to terminal.

If you decide to return the car to the Termini site, be aware that the rental returns are in a parking garage on the west side of Termini. Hertz, for example, was on the 7th floor of the garage, and while it is not confusing, it is a reeeeeeeeeeeeealy tight drive to finesse the car up the ramps without scraping one of the walls.

Dropping a car at the airport is very easy and quick. The signage to the return area is very clear and it is easy to walk from the return location to the terminal. All things been equal I would drop it at the airport in the morning around 7 or so.

In reference to wbfey’s comment.

The Garage where rental car returns are located at Termini station is the Garage Giolitti.
ES Park Giolitti Roma Termini
Via Giovanni Giolitti, 267, 00185 Roma RM
06 4470 4053
https://goo.gl/maps/JmT36DpZNKn2UjJw9

Those who cannot drive a car up a parking structure ramp without scraping walls should not rent a car in Montepulciano either (or anywhere in Italy).

I've done exactly the same thing...rented in Florence & dropped off in Rome airport 10 days later. No problems at either end. I do usually like to drop off my car the night before I fly, one less thing to consider on flight day.

Renting a car is a great way to find the places tourist buses can't, won't or don't go. But renting in Italy comes with a fair amount of anxiety. And it's not cheap. We always rent with SIXT, and take out ALL of the insurance. It means we can return a wreck and not worry.
Unless you're an outstanding driver, rent as small as possible, or expect problems.
Do the research on ZTL.
The real misery is how the speed limit changes, constantly. On ALL of the roads. And there are cameras everywhere. You're going to be watching the road, your GPS, your speedometer, and looking for speed changes and warnings. All at the same time. Sometimes the speed changes three times in half a kilometer.
We stay in airbnb's and if I have a choice, the places with "free" parking win. We've been very lucky with that.
And don't forget that you must have an International Driver's Permit (IDP).