The board at Gorton's reads "Monday, April 16th", when the scene takes place on April 9th.
When Father Gill visits the Olson home for dinner, on April 8th, there is audio of a Yankees game in the background. The Yankee season in 1962 didn't begin until April 10th.
When Jimmy Barrett's wife proposes a hidden camera TV show to star her husband, Dan suggests ABC might be interested because the network is "sick" about losing Candid Camera (to a rival network). In reality, Candid Camera had been on CBS since 1960 and hadn't aired on ABC since 1948, 14 years before scene took place.
The church handout calls the church "THE Church of the Holy Innocents," when in fact the name is "Church of the Holy Innocents."
The closing music, J. S. Bach's setting of the Lutheran Chorale tune, "Wachet Auf" wouldn't be used on Easter Sunday. The text and tune of the Lutheran Chorale are traditionally used in the Advent season prior to Christmas rather than at Easter, with its references to the second coming of Christ. Whether a Roman Catholic church in the early 60's before Vatican II would have used music based on a Lutheran Chorale (hymn) is another question.