When Eddie Brooke is put in the back of the patrol car, he is not wearing handcuffs and there's no partition between the front and back seats. This would never be considered a safe way to transport an escaped criminal.
When Barney and Andy leave the barn, they leave the radio and rifle behind. A police officer would never leave a gun unsecured like that.
According to the alignment of the actors on the set, Barney firing his gun in the barn would have killed Andy, who appears directly in his line of fire.
When Barney is left in the barn, he immediately calls Andy on the radio. While Barney responds to Andy, Andy's voice cannot be heard, as it would be in real life.
The State Police come to the office to tell Andy and Barney that they have a lead on the convict and invite them to join the search. When the officer opens the door to leave, the scenery outside the door is badly wrinkled, indicating it is a backdrop.
Right after the State Police put Eddie Brooke (the escaped felon) into the squad car and the townspeople give Barney a big head, Barney jaywalks. In a different episode, Barney gave Emma Brand a ticket for jaywalking.
When Barney puts gum into his mouth in the barn, he throws the paper onto the floor. Technically, this is littering, which is what caused the confrontation with Brooke in the first place.
When Barney comes into Andy's house, he enters from the right. There are no steps on that side of the porch for him to use.
When Andy and Barney are looking for the escaped criminal in the woods, the artificial light is visible and keeps moving around to show the actors' faces.
Thelma Lou says to Barney, "There's a killer on the loose who swore he'd get you!" It was never communicated why Brooke was in prison, so there's no way Thelma Lou would have known that Brooke was a killer.
When Andy, Aunt Bee, Opie and Thelma are inside Andy's house, they turn out the lights to surprise Barney when they think they see him arriving. Instead, it's the State Police officers who have arrived. They clearly see the lights go out and never hesitate to enter, nor question the lights going out. A real law enforcement officer would be wary of such a situation.
Both of the State Police officers are listed as Sergeants: Sergeant Miller and Sergeant Johnson. Only one of them has stripes on his uniform.
When Eddie Brooke is hiding in the loft, his pistol is several feet away. It makes no sense for an escaped criminal with a gun to not have his gun nearby.