During the free movie at the Movementarian compound, we can see the back of Mr Burns' head when the camera angle is from the back of the room. Yet, Mr Burns is later shown asking Smithers why he never told him about the cult (implying that he didn't know about them).
When Cletus pulls the shotgun on The Leader, it is clearly a break-action breech loading double barrel shotgun yet he pumps it like a pump-action shotgun (with associated sound).
Burns remarks that "we're getting screwed!" Several other episodes have shown Burns's ignorance of modern colloquialisms.