Continuity mistake: In the episode with Phil Silvers as Harold Hecuba (a.k.a. H.H.), H.H. goes out to change costumes. When he comes in as Ophelia the stage door first closes behind him, then falls off the hinges. The shot cuts to the Castaways, then back to H.H., and the door is back on the hinges.
Plot hole: The scientist yells for Igor to capture the escaping Castaways, to which Igor responds "capture them yourself" in Ginger's voice. The scientist appears surprised to find out Igor is not Igor, but how can this be? He had to be the one to pull the level to switch everyone's personalities and bodies around, so he should have already known.
Visible crew/equipment: Toward the end, the group is around the table, talking about Gilligan. Then they see a pear and a bunch of grapes float over to a plate at the head of the table. Skipper goes over and addresses the invisible Gilligan. If you look at the top of the hut, you can see the wires used to make the fruit float. Also, earlier in the hut with Ginger, when Gilligan bit the pear, you can see the wire holding the pear as it shakes after the bite.
Audio problem: Towards the end of the episode, the castaways read Tongo's note that he can't take them back with him due to how he behaved with the gorilla. Mr. Howell yells "we won't tell anyone!" followed by other random comments from the others for Tongo to come back. Tongo leaves and as the camera pulls away from the castaways, the same audio repeats itself even though this time doesn't match what anyone is saying.
Plot hole: This was obviously bad script writing, but let's figure out carefully how this failed rescue attempt (by Gilligan) doesn't really make sense. The two Russians were to be picked up at 11:00. When the castaways appear at the Russians' location, the Russians are gone. Gilligan states it's 8:00 according to his watch, due to the unaware fact he set his watch to Manila time (3 hours earlier). He claims to have set his watch when it was 10:30 in Manila - Skipper then yells there's 3 hrs difference between the island and Manila, so Gilligan sets his watch ahead 3 hrs and it's now 11:00. Here's the mistake: if it was 10:30 in Manila and there's a 3 hr. difference, it would have been 1:30 on the island, and also if Gilligan's watch stated it's 8:00 now (Manila time), he would have set it to 7:30 when hearing the 10:30 announcement on Manila radio (which he didn't do). Lots of these times just don't make sense.
Revealing mistake: Near the end of the episode, all the castaways are passed out cold when Lord Beasley finally catches his butterfly. Right after saying the line "That's the trouble with you Americans - you simply don't know how to hold your tea" as he's exiting off camera, Mrs. Howell starts to open her eyes slightly, but then quickly shuts them again. Perhaps she thought the cameras stopped rolling.
Character mistake: When the "Honey Bees" (Mary Ann, Ginger, Mrs. Howell) perform, Mrs. Howell (Natalie Schafer) messes up her lip-syncing throughout the number. But it is most noticeable at the very end when the girls put their heads together and say "buzzzzzzzzzzz" but Mrs. Howell is mouthing an "oooooooooo" which she changes to a "zzzzz" mid-way through the note.