Continuity mistake: When the Pope sneaks up behind Cardinal Rocco in the basement, in some shots the Cardinal has a drink in his right hand, in others he doesn't.
Audio problem: When Robbie is playing his guitar for the children early on in the film, the way he is playing simply wouldn't sound like what you hear - it's very badly dubbed.
Factual error: When Robbie is shot at in the "Popemobile", the bullet comes through the glass. The whole reason for having it is because it's bulletproof. They started using it after John Paul II was shot in the early 80's.
Audio problem: When Luccia is playing the piano, it's some of the worst miming and dubbing I have ever seen. She is pretty much at totally the wrong end of the keyboard, and you frequently hear trills, when all she's doing is playing chords.
Revealing mistake: When the rock band drop Luccia off at home, they knock over a statue. It's supposed to be stone or marble, but you can see by how much it moves and how it falls over, that it's actually plywood or some other lightweight material.
Continuity mistake: There is no way the uniform Robbie Coltrane steals from the Swiss Guard would fit him, yet it is seen to be very loose fitting.
Factual error: The real "Popemobile" is famously a Mercedes-Benz, the one in the film isn't. Whilst the film makes numerous factual errors for comedic purposes, it's very unlikely that this was intentional.