Jen, 'siesta' in Italian cities is largely a thing of the past although it's not unheard of for owners of smaller shops to head home for a late lunch. Many but not all restaurants can close for a few hours between mid afternoon and opening evening hours.
Major attractions do not close. Churches are all different; some close for a few hours midday into the early or mid afternoon and some don't. A few also close to visitors on certain days. Visiting for sightseeing purposes is discouraged during open services so heavy mass rotations at some - Santa Maria Maggiore, for instance - makes pre-planning a necessity!
As this is a Jubilee Year, best to consult the official calendar for any special services which may interfere with usual visiting hours as the 'major' basilicas or others of the larger churches.
http://www.iubilaeummisericordiae.va/content/gdm/en.html