Florence has a wonderful buzz about it at Christmas. You'll have no shortage of restaurant options on Christmas Eve, however Christmas Day (and to a lesser extent, the 26th, St. Stephens Day) will find only a sparse number of restaurants open for business. A couple of ideas/suggestions for your visit: Check to see if the (Santa Croce) German Christmas Market is still open by your arrival on the 20th; that's usually right around the time they're packing stuff up and closing down the stands [it runs off the Advent Calendar, so the event's exact dates change annually, and I'm not sure when they fall this December]. Back to restaurants, we really enjoyed a Christmas Eve dinner at Da l' Conte Diladdarno. The food and service couldn't have been better ('memorable' in fact) and, as it was the night before Christmas, the cook, the owner and the waitstaff all took turns coming by every table to share holiday wishes and offer extra plates of goodies. It was like being in someone's home. Start with their 'cutting board' (tagliera) and let the meal evolve from there. For Christmas Day, other than hotels, there aren't many restaurants open. We reserved a table months' ahead at Trattoria 4 Leoni for the traditional, mid-afternoon seating on Christmas Day, and the food was meh... If you really want a restaurant dinner on Christmas you'll need to book a table well in advance for the few open establishments.
Two other food related recommendations for your visit: Trattoria Sostanza's 'butter chicken' (pollo al burro) is a must; and near the Uffizi Gallery is a butcher shop called Alimentari Uffizi. In between waiting on locals and fulfilling their to-go orders, the butchers there have three small tables and will create for you a wonderful assortment of focaccia and olives, cheeses and prosciutto, with a little locally made vino. They're open on the 26th (that's when we went), so a good lunch option on St. Stephens Day.
One last restaurant recommendation (occurred to me while typing) would be Al Tranvai, another small intimate place across the river and open only for dinner. It was so good that after eating there we cancelled two other (nights') reservations and returned for a total of three dinners. Same for Da l' Conte Diladdarno (Christmas Eve recommendation) ... we had to return a second time in lieu of other pre-made reservations, it was so good.
None of the recommended places are expensive nor what I would call 'fancy', just good food served by nice people. Have a great trip!