Factual error: When they are riding the life raft off of the top of the snow-covered mountain watch how it falls. All three people were close to the front of the raft, yet the raft drifts backwards before plunging in nose first. That would be impossible with all of the weight shifted to the front of the raft to begin with. (00:18:25)
Other mistake: In the "tunnel of bugs" you may reasonably expect there to be such creatures as cockroaches, spiders or giant centipedes, but the majority of the insects are locusts which are herbivorous and would not be found in an underground tunnel. Other out-of-placers are giant stick insects (also herbivorous) as well as a South American harlequin beetle (this part of the film is set in India). You can also see, in the scene where Willie turns her hand over to reveal one of the giant stick insects, that she is cupping it to hold the insect in place.
Plot hole: Mola Ram has his men knock over a vat of water in a last ditch effort to kill our heroes. First of all the vat can't be that big if it can be knocked down with a few sledgehammers. Secondly, for the amount of water that chases Indy and co. down the vat would have to be big enough to fill the room it's in, which obviously it's not. Third, Indy and co. travel through several large caverns during the mine cart chase, caverns that could easily swallow up any of the water flowing through the tunnels so there shouldn't be any water left to catch up to Indy by the time they stop.
Continuity mistake: While Indy and the Thuggee Chief Guard are battling it out, after the shot of the Maharaja gleefully stabbing the Indy doll, in Willie's next closeup she's wearing the veil as she throws rocks at the Chief Guard, but in the next wideshot the veil is gone, but it's back on her head in following shots.Super Grover