This is a good starter list published by the National Catholic Register for the Year of the Priest – though I think it leaves off some very good other films about priests. For some reason they are staying with U.S. films and it does not include films about bishops or popes. The list does bring in DIARY OF A COUNTRY PRIEST, directed by Robert Bresson.
The Bing Crosby films may be classics but I think they present a very romanticized vision of the priesthood. KEYS OF THE KINGDOM, the book and the film, predate Vatican II, as does DIARY, but both breathe the spirit of the council.
I would add:
Keeping the Faith , The Third Miracle, Becket (OK he became the Archbishop of Canterbury…). There are challenging films as well: True Confessions, John Sayles’ MEN WITH GUNS (one of the story lines is about a priest), El CRIMEN DE PADRE AMARO, Graham Greene’s THE POWER AND THE GLORY (1933; Preston Sturges wrote the screenplay); ROMERO (there was the archbishop as well as the priests who worked with him; now available on DVD).
Feel free to add your favorite film by commenting.