Barney's newspaper is at first clearly unblemished, but in the shot just before he rips it in two, a tear can be seen in the top crease that wasn't there before, enabling him to rip it evenly.
In the scene with the mayor, the phone's extension cord hangs in front of the sheriff's desk drawer, wherein lies Opie's fish. During the sequence with the fish, the cord suddenly hangs over the side, out of the way of the drawer, but it returns to hang over the drawer again for the following sequence involving the crying baby.
After Barney blows a kiss on the phone, he puts the phone down on the sheriff's desk. The scene cuts to a view from the front door and the phone now sits in a different location in relationship to the gavel. The phone returns to its previous location (apparently on its own) by the time it rings, after Barney hands the files to the mayor.
The office mirror is present and missing in some scenes.
The bear that climbs into the sheriff's car is a Sun Bear (as identified by its tan "necklace"). These bears are native to Southeast Asia, not North Carolina.
The City Mayor would have no authority over the County Sheriff.
When Barney sees the mayor through the window, headed toward the courthouse, he runs to get back in the cell. When he slams the cell door, all of the bars, as well as the brick wall next to the cell, wiggle revealing them to be a stage prop.
Barney can still be heard whistling when the mayor and Andy enter the courthouse, but his mouth is clearly in no form to actually make a whistling noise.