Continuity mistake: On the mine conveyor, the guard pins Indy to the belt with his foot and his head toward the grinder. When Short Round tackles the Maharajah, we can see the guard on the conveyor, but he's facing the wrong way, both his feet are on the belt, and Indy is nowhere in sight. (01:31:30)
Other mistake: During the Tri-Motor crash sequence, the crew dumps the fuel from the aircraft and bails out. In the shot of the red fuel pouring out, the fuel is shown streaming out of the exhaust pipe for the center engine, which would be hot enough to cause the fuel to ignite and either set the aircraft on fire or cause it to explode.
Continuity mistake: During the mine cart scene, Indy hits the second lever with the shovel to change the course. The Thuggees in the cart behind him shoot the same lever and it flies back just as Indy hits it, and changes the course again.
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)
Plot hole: In Raiders of the Lost Ark, Indy tells Marcus he doesn't believe in magic or superstition. However, in this movie, which is set a year before Raiders, Indy not only witnesses supernatural acts, but takes advantage of them to win the day (by using the incantation to ignite the stones at the end). Therefore, by Raiders Indy would not only believe in magic and superstition but would know they exist. Listed for this movie as it was made after Raiders.
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.
Other mistake: When fighting the sword wielding guards before getting on the bridge, Indy uses his bullwhip to disarm one of them. As the sword flies out of his hand it kind of hangs in the air as it falls. It is obviously a prop made of a very light material. (01:42:35)
Factual error: The image of the plane's instrument panel shows a VOR, just to the left of the orange circle with the triangle in it. The VOR is "a type of short-range radio navigation system for aircraft, enabling aircraft with a receiving unit to determine its position and stay on course by receiving radio signals transmitted by a network of fixed ground radio beacons. ... Developed in the United States beginning in 1937 and deployed by 1946..." (Wikipedia). The movie's opening scene is Shanghai, 1935. (00:16:45)