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.
Continuity mistake: When Indiana is fighting on top of the tank, he kicks the periscope. It spins counter-clockwise, but the handles inside spin clockwise. (01:32:45)
Continuity mistake: Indy pours water from the grail into his father's mouth. Between two consecutive shots, the position of Indy's hand changes and the direction he pours changes 90 degrees.
Continuity mistake: The shape of the exploded canon barrel on the tank changes a lot in the different shots. Especially between the scene right after the explosion and the scene where Indy is hanging under the barrel.
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.
Continuity mistake: When Indy meets Hitler, there is a shot over Indy's shoulder that shows his lips are slightly parted (relaxed). Cut to Indy's face and he closes his mouth tightly. In the very next shot, with no elapsed time, we see from Indy's shoulder again that Indy's mouth is relaxed as before. (01:10:15)

Other mistake: While Indy and Henry are aboard the zeppelin Indy tosses Vogel out the lounge window, and when Vogel shouts up to the departing airship, in the next exterior shot we see the zeppelin airborne against a scenic view. Later, when Indy realises the zeppelin is turning around it cuts to another exterior shot of the airship, and the scenic view is identical to the earlier shot, only it's been flipped. The images of the zeppelin facing two different directions were composited onto the same scenic background.
Factual error: When Indy and Marcus first arrive in Venice on the boat the dock they alight on to has tactile pavers (for the visually impaired) installed on the platform. These weren't invented until 1965, some 25+ years after the scene takes place.
Suggested correction: One piece of the pier looking different doesn't mean that is meant for blind people, could be just a repair, or decorative, or a coincidence, could be anything.
You can see it is a tactile paver as it has the raised spots.
Continuity mistake: The Zippo lighter on the floor, thrown by Dr. Jones, keeps moving around between shots.
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.
Audio problem: When Vogel is holding Elsa hostage in the castle, he points his gun at Elsa's neck and says "Enough, she dies." If you look at Indy's shadow on the wall, you can see Indy is mouthing the word "Wait." In the next shot, he is saying the word again, this time with audio intact.
Visible crew/equipment: After Indy's father is cured and the Nazis drop the weapons, he stands up and set lights are reflected on his glasses.
Continuity mistake: When Sallah pulls his rifle and overtakes the Nazis at the Great Seal he is in the forefront with the Nazis in front of him and Indy and his father are behind him. Within the next scene he is now in the background with his rifle resting in his arms.
Continuity mistake: When Indy is pouring the water from the Grail onto his father's wound, we see him pour all the water out. When his father takes the Grail, there's now water left inside.
Suggested correction: Automatic refilling seems to be the least of the miracles the Grail has performed.
There was no evidence of this, plus we never see it refill before or after. How would it be empty when they first find it if it miraculously refilled itself? This is a poor correction just to make a correction.
Continuity mistake: Onboard the ship at the end of the prologue, after Indy grabs the cross before it falls over the side, the guy in the white suit climbing up the steps. He's only reaching the top as the shot changes. In the next shot, he's already a few steps away from the top of the steps, telling his men to stop Indy.

Continuity mistake: During the temple quake sequence towards the end of the film, Elsa fails to maintain her grip on the rising earth and falls onto tilted ground, sliding towards the chasm. At this point, Sallah is kneeling on the ground, but in the very next shot, he is now grabbing hold of Henry.
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: When Sallah rides up to the tank to help, just as Indy shouts, "Get Dad!" in the closeup and wideshot of Henry we can see that he's actually lying on protective support, and not lying directly on the moving tank treads. (01:35:40)

Visible crew/equipment: When Indy tries to reach the Grail his father convinces him to "Let it go" and Henry pulls his son to safety, then in the next wideshot of Indy, Henry, Sallah, and Marcus a crewmember wearing a short sleeved, white shirt and blue pants can be seen at the right side of the screen, just before it cuts to Henry's closeup. This crewmember can be seen again after the closeup of the knight. (01:57:45)