In pure theory, the local policemen (Polizia Municipale) might ask you if you have a permission to take pictures with a tripod. They refer to old legislation which is mainly applied only to cine activity: people needing to do footage do ask permission, and pay a fee, but also have police assistance in a number of ways, and not only with parking (film director Nanni Moretti famously had prostitutes "moved away" by the police when he was making a film).
A "sane" and normal local policeman will not bother you about that, but that does not imply that all of them are sane, or normal.
That said, you will be quite immediately stopped from using a tripod in Piazza del Quirinale, for (idiot) security reasons. Reason given to me: a tripod can be used to sustain a rifle. Strangely enough, you will not be stopped in Piazza di Montecitorio or in Piazza Colonna. Oh, also in front of the Synagogue a tripod will have a policeman call you although they might tell you that you can take the picture, but they have to register your documents, that's my experience at least.
Other than that, you cannot obstruct pedestrian movements in crowded pavements, let's say Via del Corso during the afternoon.
Finally, sometimes some private security guard will come near you to ask you what you are doing. You can just ignore them but remember they are just working and more or less trying to guard their ass and secure the soup, 30 seconds chit-chat and they are fine. They know they cannot prevent you from taking pictures, they just want to show to their boss that they are actually working, inspecting potentially suspicious situations, that kind of theatre.
Often you will be told you cannot use tripods inside churches. In that case, the bean bag is the way to go. The reason for this is that old historical churches belong to the State by a fund managed by the Interior Ministry, and they reserve certain rights not just about taking pictures but also about using them in publication. Although the norms are there, it's all seldom and randomly enforced, you might as well find that nobody tells you anything about tripods in churches.
Musea will get very, very pissed if you open a tripod instead.