Updated with new links to the PDF map on December 3, 2011:

The list below is from the new Florence Municipal website in English - I wouldn't make definite plans by it though, as at least one of the locations (Loggia del Grano) has been closed for at least a year. I know for sure that the locations at Borgo Santa Croce and Via dello Sprone are open. The Piazza Ciompi location is inside the small garden across from the main square/flea market. We will be making a map of these locations soon. Note that most locations have a charge of .60 euro cents.

Stazione FS S.Maria Novella - underpass
hours: 08-20
opening time: all year round
Euro 0,60

Via della Stufa, 25
hours:  10-20*     9-22**
opening time: 
* from 01/11 to 31/03
** from 1/04 to 31/10
Euro 0,60

Via Filippina on the corner of Via Borgognona
hours:  10-20*     09-22**
opening time:
* from 01/11 to 31/03
** from 1/04 to 31/10
Euro 0,60

Borgo Santa Croce, 29r (very near to the church and square of Santa Croce)
hours:  10-20*     9-22**
opening time:
* from 01/11 to 31/03
** from 1/04 to 31/10
Euro 0,60

Piazza Ghiberti - inside the S.Ambrogio market
hours: 07-14
opening time: all year round
closed on Sundays and holydays
Euro 0,60

Via dell'Ariento, 14 - Mercato Centrale (Central Market)
hours: 07-14*     14-19.30**
opening time: all year round
* inside the marketplace
** outside the marketplace
Euro 0,60

Piazzale Michelangiolo - Viale Galilei
hours:  10-22
Euro 0,60

Via dello Sprone
hours:  10-20*     9-22**    
opening time:
* from 01/11 al - to 31/03
** from 1/04 al - to 31/10
Euro 0,60

Fortezza da Basso - inside the garden
 
Lungarno Pecori Giraldi - Tourist bus terminal - inside the multiethnic area
hours: 10-17*     09-19**     09-22***
opening time:
* from 01/10 to 20/03
** from 21/03 to 31/05
*** from 01/06 to 30/09
Euro 0,60

Via Sant’Agostino, 8
hours: 10-13*     15-19*     09-14**
opening time: all year round
closed on Wednesdays and on holidays 
* on Monday, on Tuesday, on Thursday, on Friday
** on Saturday
Euro 0,60 WC
Euro 1,81 shower

Piazza Madonna della Neve
hours: 7-23* 7-02**
opening time
* from 1/11 to 31/3
** from 1/4 to 31/10
Euro 0,50
 
Piazza de' Ciompi (inside the garden)
hours: 9,30-17,30* 9-19,30**
opening time
* from 1/11 to 31/3
**from 1/4 to 31/10
free

Loggia del Grano (behind the Uffizi - but pretty sure this location is closed!)
hours: 10-18
opening time
all year round
Euro 0,60
 
Parco delle Cascine (lots of construction and renovation going on around here - would not count on this being open all the time)
hours: 10-20* 9-222**
opening time
* from 1/11 to 31/3
**from 1/4 to 31/10
Euro 0,60