Continuity mistake: When the Hatay soldier is decapitated in an attempt to navigate the temple's defences, his head rolls along the ground and Elsa turns her head around and looks over her shoulder. In the next shot, she has now turned her full body around, and in the shot after that, she is looking over her shoulder again.
Continuity mistake: When Elsa's hand slips from its glove whilst Indy is holding on to her, Elsa falls and she is holding the glove in her right hand, but when she falls into the chasm, the glove has gone.
Revealing mistake: During the castle scene, when Indy is swinging outside, the sky is clear and blue and obviously not rainy, yet it is still raining. And yes, sometimes it can rain when it's sunny, but take a look - it is NOT rainy weather.
Revealing mistake: The stuntman who saves Indy's dad from the tank has a perfectly trimmed beard and a thinner face than Sallah's.
Continuity mistake: When Elsa is looking out over the Canyon of the Crescent Moon, the sun is low - it is close to sunset. However when Indy is riding through the very narrow canyon, the sunlight reaches the bottom. It is not the same time of day (or later the same day).
Revealing mistake: After the Nazi tank drives off of the cliff and hits the bottom, the upper turret breaks off. As the turret rolls over, you can see there is no hole in the bottom, just in the top.
Revealing mistake: When Indy and Kazim are fighting on the front of the boat, the front of the boat doesn't move at all when the huge propeller hits the rear of the boat.
Factual error: When Indy and Marcus fly to Venice, the route map shows them going from New York to Botwood, Newfoundland, then from Botwood to the Azores to the continent. There were two transcontinental air routes at the time across the North Atlantic: one from New Work to Botwood to London, and one from New York to the Azores to the continent, but none from Botwood to the Azores. (http://people.hofstra.edu/geotrans/eng/ch3en/conc3en/earlyairnetworks.html).

Revealing mistake: When Elsa falls through the crack in the hole, she is swallowed by a big cloud of smoke in the void. However, her silhouette is visible underneath the smoke, revealing she never falls any deeper, and thus giving away how shallow the hole was.
Continuity mistake: When Senior is rescued from the tank with a whip around his ankle, the whip dissappears a split second later, when he is being mounted on the horse.

Continuity mistake: When they are first looking over the cliff at the tank, Sean Connery's beard is quite full, and in a shot moments later, still looking off the cliff, it has been trimmed. This is due to the fact that this scene was filmed after principal photography had ended.
Revealing mistake: When Indy is clinging to the side of the tank turret in the desert, to the right of the tank are tire tracks from the vehicle shooting the scene. They appear to just start at a set point. There was likely no way to hide this.
Suggested correction: There's no obvious moment in the film where tire tracks are visible while he's clinging to the side. A specific time or shot description is needed.
Visible crew/equipment: By the cliff, Indy's hat comes rolling towards him. Watch closely and you'll see a long shadow a meter above the ground, parallel to that of the crew member throwing the hat.
Other mistake: In the scene with the tanks, Indy rides up beside one with a rock. Watch the gun closely. The end of the one that he is supposed to shove the rock into is wobbling inconsistantly with the rest of the gun, or the tank for that matter.
Revealing mistake: When the VW gets stuck on the front of the tank and is shot off, you can see that the car has no engine, no body in the car, and no clutch. Also, when the car is blown away you can see a wire used for the stunt fall right in front of the camera.
Continuity mistake: The back of the boat gets chewed up by the prop in the same place twice.
Revealing mistake: When Henry says "Faster, boy, faster!" to Indy when they are driving into the tunnel to escape the flaming aircraft, you can see the track that the flaming aircraft is sliding on.
Revealing mistake: When Indy shoots three Nazi soldiers with a single bullet during the tank chase sequence, you can see the blood packets inside their uniforms.

Revealing mistake: When the car is chased by a plane and enters a cave, check the rocks above the entrance and you'll notice that both the entrance and exit are the same location and the final shot has been flipped.
Factual error: When the soldier from Hatay attempts to pass through the booby trap and his head is chopped off there is no blood. Any wound to the head or neck tends to bleed profusely but the head rolls by without a drop.