Gas stations in Italy don't really close. Except for the very small ones, you can still refuel at any time using the self service pump. When they are "closed" (as you say), they are just totally unattended but you can still refuel. Therefore on Sunday, lunch time (1-3:30), and nights, you need to use the automatic payment machine to insert money or credit/debit cards (chip and pin only) because there will be no cashier either. Just insert the money (or credit card) in the payment kiosk, push the number of the pump you want to use, and refuel. I have refueled on Sunday and during "closing" times twice this month, including just before returning my rental in Florence (it was lunch time break).
Sunday may not be a good day to take trips for other reasons, not so much for fear of being stranded without fuel. Buses are on a limited schedule, some shops are closed, some museums may be closed, and of course some popular spots (like Cinque Terre or San Gimignano) may be more crowded. Also most car rental agencies, except for airports, may be closed on Sat afternoon and all day Sunday.
Wednesday's and Tuesday's may be better for air travel, compared to weekends.
Some museums close on Moday. You just need to check the opening hours online and plan accordingly.