As a rule, you cannot use flash in museums and churches, that’s a pretty universal rule, not just for Italy. Additionally, many museums and churches now have a “no camera” policy, so if you go to a museum or a church, try and find out in advance whether you will be able to take pictures or not.
In Italian churches, look for a sign that will tell you taking photos is not allowed.
I had to do a little research on this one. There is a very strict rules against cameras inside the Parliament buildings in Piazza Montecitorio.
There is absolutely no photography or video taping allowed in the Sistine Chapel, just so you know… The no photo and no video rules are strictly enforced there, and the guards pay very close attention to the visitors. While in most museums the reason for no photos is so that the flash would not affect the pieces of art, the reason for no photo and no video are a bit different for the Sistine chapel. Nippon TV has exclusive rights to photo and video there, due to funding the restoration of Michelangelo’s art in the chapel.
Also, in most places where no photo is allowed, you can buy post cards that are quite nice.