While Batman and Robin are climbing the wall of the Last Longer Warehouse, just after they stop to talk, the string holding Robin's cape up suddenly either breaks or comes loose and his cape immediately falls down (towards the wall), revealing they are not actually climbing upwards.
It's quite obvious there is a stunt double for the Joker in the fight scenes with Batman and Robin.
Under strong lighting, it is apparent that The Joker's wig is a fine mix of brown and green hair. When the camera is pointed at a television screen showing the Joker (rather than directly at the actor) the wig appears all brown, likely a glitch in the early development of color television.
The Joker's vintage French champagne bottle is variously dated to both 1849 and 1949.
The Joker's hair appears brown when he takes over television. Although not necessarily a continuity error (considering character was never confirmed to have permanently green hair), it is an odd occurrence considering he is almost never seen with other color in the series, unless wearing a disguise.