Hi Lisa,
Hopefully a few others can chime in with their experiences, but from what I've heard/witnessed, it depends on the church. Generally, if the mass is closed to tourists, someone will be posted at the door, but will let you in if you say you are there for mass. It can help to have a letter from your priest (assuming you are Catholic).
Other churches are far more relaxed. We (heretical nonbelievers) stumbled by accident into an English mass in Santa Maria in Trastevere. We took a seat in the back and decided to stay and observe the service, taking our photos afterwards. No one seemed to mind, and we saw tourists and faithful alike wandering in and out during the entire thing. Might have helped that it was a Thursday evening instead of Sunday.