From shot to shot the priest's cross changes position. It's an orthodox cross with a slanted second cross piece. It flips position randomly.
Towards the end of the film, Tevye says his daughter's plea, "My people are your people," is from the Book of Ecclesiastes. It's actually from the Book of Ruth.
When he recites the Havdalah prayer ending the Sabbath, Tevye forgets to include the blessing over the spices.