Plot hole: Many episodes show either visitors on the island swimming away or objects floating away on rafts (a lion, a silent film, etc) that at the end of the episode always make it safely to civilization (as heard on the radio)... yet, none of the castaways ever could swim or leave on a raft to attempt to get off the island and get rescued in the same way.
Plot hole: On many different occasions, the castaways meet a wide variety of people who say that they have been on the island for either ten or more years thinking they were the only ones on the island before the SS Minnow even arrived. With so many people who had been on the island for such an extended time, it seems kind of strange that none of them had ever met one another.
Revealing mistake: When the Professor comes over to the Robot to ask "How would you like to do us a favor?", look closely at the mesh like part of the Robot's "face", just left of the vertical black bar. You can see the Robot actor's eye blinking and looking around. When he nods, you can see the outline of the actor's head.