Your plan is not crazy enough. But I'm crazier than you and I'm from Tuscany, so here is your truly crazy plan.
Depart the Cinque Terre (you don't say which village) as early as you can, take a train to Aulla (in the Lunigiana area) near La Spezia..
Use the Trenitalia website below for train timetable. You will have to change trains at La Spezia Centrale.
http://www.trenitalia.com/
Once in Aulla you will take the famous scenic train to Lucca through the Garfagnana valley (it's a historic scenic rail line). Note that the steam trains are used only on special occasions. Generally regular commuter trains are used. This year the line started to be served by the modern Swing commuter train.
The Aulla-Lucca timetable is below:
http://www.ferrovia-lucca-aulla.com/orari.html
After the amazingly beautiful train ride, alight the train at Lucca. There is no luggage storage inside the station, but the tourist center in the Station piazza does offer that baggage storage service (details below, under "services").
http://www.touristcenterlucca.com/index.php?lang=en#
Once you are done with Lucca, take the train to Pisa. Use the trenitalia website a eve for timetables. Pisa Centrale has a luggage storage. If you get off the train earlier at the Pisa San Rossore station, you are walking distance from the tower, but there is no storage for your luggage. The website for the luggage store at Pisa Centrale is below. They close at 21:00 hours, so make sure you retrieve your luggage before. There are trains to Florence until well past 10pm. One hour later you are in Florence.
http://www.depositobagaglipisa.it/