Deliberate mistake: During scenes at Walker's Drugstore there are many magazines by the wall near the door, which have tape/paint hiding part or all of their cover titles. Examples are: all the Sports Illustrated have tape across the word "Illustrated", the House and Garden has its "House &" covered, and Time magazine's entire title is blacked out.
Continuity mistake: At the "Hen Party", Jeff Pruitt says hello to Thelma Lou and she states she is going to get coffee, then goes to the kitchen. However, when Jeff walks across the room and looks back, Thelma Lou is back in the same area. She never walked back from the kitchen.
Continuity mistake: Scenes shot at Andy's porch have significant differences depending on the filming location, as there are two sets of Andy's porch. One is just the facade of the house in the 'town of Mayberry' in the studio's 40 Acre backlot; and the second is located in the studio's soundstage. Note in some shots the door's sidelights have vertical muntin bars, the screen door has a white handle, whereas in other shots the sidelights' muntin bars are gone and the screen door handle is black (screen door's gone season 5). Also, note the significant difference in the width of the wall between the parlor windows and the door, the height of the parlor windows from the floor, etc.
Plot hole: In the episode where Barney confronts the two farmers for selling produce on the side of the road, Barney mentions that they're not allowed to sell because of "section 17.4" of the ordinance code - that street vending is not allowed, and Andy confirms it. but in the episode The Merchant of Mayberry (I think), Bert comes to town and is selling dry goods, etc. and he's allowed to sell. Even Ben states that there must be a structure if he's selling - wouldn't you think that Andy and Barney would have known that already before setting him up to sell?