In the long scene in Andy's office, when Ed Sawyer meets Lucy Matthews, the steam heat radiator and the spittoon on the floor, beneath the bulletin board, change positions along the wall at least four times during the scene. They keep moving back and forth between the glass-fronted bookshelf on the left and the door to the back room on the right.
A steam radiator is visible in Andy's office just to the left of the back room doorway. By season 2, it's been replaced by a small furnace. By season 5, the radiator has been replaced by a vent that spews black smoke when Newton Monroe (Don Rickles) fails to open the flue to the basement furnace.
At the hotel, Andy holds his right arm across his abdomen, but as the scene changes to a wider shot, his right arm extends across the counter top.
While cutting Barney's hair, Floyd rambles on about what he would do if he were running government affairs "down in Washington". Mayberry is located in North Carolina, far south of Washington D.C., so Floyd should have said, "up in Washington".
The courthouse back room calendar shows a month with 31 days, with the 6th on a Sunday (putting the 1st on a Tuesday). Showing the next month to have only 28 days (and the previous month 31 days), that would make the month January and the only January to have the 1st on Tuesday in a non-leap year up to that time was 1957. The next would come in 1963. EDIT: However, as this is a "back room", it just may be a neglected task to replace the calendar, no one may care that it's an old calendar or they may simply be using a month from the old calendar that aligns with the current month, rather than buying a new calendar. (People were thrifty in that era.) Regardless, it does not meet the criteria for a "Revealing Mistake".
The two customers in the barber shop are looking out the window at the stranger getting off the bus with Floyd the Barber, Andy and Barney behind them. The video plays forward for a small amount of time and then it plays backward before the shot cuts to the reverse angle.
Andy is in the barber shop reading the paper and as he is reading about Joe MacKnight celebrating his 103rd birthday, the top left-hand side of the paper opens up. It reveals that the paper he is reading from, "Local News", is actually The New York Times. "York" can be seen, as well as the "T" in Times, plus the distinctive "New York Times" font, which is copyrighted by the New York Times.
Jason the hotel clerk comes out of his office as Ed is checking in and shuts the door behind him. But the door pops open again even though the sound of it latching can be heard. It's obviously not a real door. The door stays open throughout the scene, but the size of the opening varies between shots.