The weather is not really bad in Tuscany during the winter, it primarily gets cold and rainy. It can snow in the mountains, but that at the higher altitudes.
Do you already know where you'll be staying? The area along the coast is the part that is most seasonal in nature, when the beaches attract many visitors during the summer and pretty much are sleepy towns in the winter, with many shops closed. But all of the main cities are open year round and are perfect in the off season when crowds are virtually non-existent. Aside from seeing some gelato shops take vacation in January or February, I would say pretty much stays open in the regular tourism destinations - Florence, Pisa, Lucca, San Gimignano, Siena... the towns in Chianti and Val d'Orcia are definitely quieter but still "open".