When Robin picks up the suitcase on the side of the road next to the gates of Lord Ffogg's mansion he undoes the right hand latch then decides against undoing the other latch and places the suitcase on the ground without re-closing the latch. When Batman arrives on the scene a close up of the suitcase shows both latches closed. Batman then undoes the right hand latch.
When the "bee" lands on Robin's hand you can see the wire that pulls it from the hive to his glove.
After Robin cuts the mooring line to keep the girls from getting on the school boat, the footage showing the boat "drifting" away clearly shows a boat under power. Because drifting vessels are being carried by the flow of the water they very rarely leave a wake, even if they are drifting bow first.
When Batman repeats the Indian rope trick outdoors, the string pulling up his rope is clearly visible.
Batman (Adam West, who had poor peripheral vision when wearing the Batman Cowl mask) bumps his head against the archway leading to Lord Ffogg's dungeon.