In Tim's upstairs garage apartment there's a cone fireplace. Sometimes it's freestanding in the middle of the living room and in other episodes it's in the corner with a brick surround.
The first seven episodes of the first season showed a copyright date of MCMXLIII (1943) instead of MCMLXIII (1963). This was corrected in episode eight.
Many times throughout the series, when Martin's antennas are coming out of his head, with close inspection, the antenna isn't right on his head.
Throughout most of the first season, the wires supporting various "levitated" objects are obviously visible.
In at least two episodes, in the scenes where the antenna is to raise out of Martin's head, the stagehand (whom is to do the effect) can be spotted.