Continuity mistake: The professor mentions he can't drink vodka with the Russians because he's allergic to alcohol, yet in the episode "Man With A Net" (with Lord Beesley), he drinks a strong alchoholic drink he made along with everyone else with no signs of any sickness, or reaction preventing him from drinking it.
Plot hole: Gilligan admits he put the Professor's measuring stick in deeper and deeper waters to catch bigger lobster. So wouldn't the Professor have noticed the stick has been moved further out (and more than once too) in the water rather than think the island is sinking? He always had all positions and locations down to a science in many episodes, so how could this have gotten past him?
Other mistake: At the end, the Professor and Skipper walk up on Gilligan, who's busy chiseling a message on a stone. Look carefully, because he's hitting the chisel, but the tip never touches the stone, and the 'E' in 'HOME SWEET HOME' (the message Gilligan is working on) is plainly visible.
Revealing mistake: After the Skipper throws the attache case in the lagoon, it explodes. Problem is that the lagoon water and the water with the explosion are in two different areas, and where the explosion is, there was no case floating. Just markers floating where the explosive is placed underwater.
Other mistake: At the end, after Mary Ann puts the pie down, and Gilligan drinks the juice, he hits the table, flipping the pie into the skipper's face. First, the Skipper starts to wince when Gilligan hits the table, preparing for the gag. Second, the pie tin defies physics by staying exactly in place on the board after the pie leaves it, and after the board bounces once and settles down. Third, Gilligan dips his finger directly into the cream on the Skipper's face, rather than missing a few times.