Continuity mistake: During the middle of the bar fight scene, Indy was slammed on the counter and whispered to the girl, "Whisky". In the next shot she grabbed the bottle and handed it to Jones. The very next shot she hands him the bottle again. It happens very quick. Watch carefully. (00:32:05)

Continuity mistake: In the market chase scene Marion is confronted by a guy with a knife. In some shots the sharp edge is facing upwards, in others it's downwards. (00:39:10)

Continuity mistake: When Indy is trying to stop the truck that is supposedly carrying Marion in a basket, Indy shoots the driver as the truck heads towards a "dead end". The driver's foot hits the accelerator and the next shot opens up to a big open space where the truck tips over and explodes. The "dead end" has disappeared. (00:40:10)

Continuity mistake: When Indiana is confronted by the Cairo swordsman, he already has his left shoulder pointing towards his opponent in the shot where the crowd scatters. But in the following shot, where the swordsman is first seen, Indiana has his right shoulder pointing forward and is just turning around to his left side. (00:40:14)

Continuity mistake: When Indy is running through the streets of Cairo looking for the men carrying Marion in a basket, the sweat stain on the back of his shirt shrinks and grows. (00:40:35)
Continuity mistake: When Indy is entering Marhala Bar, Steve Hanson is right behind him, following Indy inside. In the next shot, Steve is suddenly standing next to Belloq (he seems to have arrived before Indy did). (00:43:08)

Continuity mistake: When Indy is deciphering the object with the old man, there is a telescope behind them. It changes orientation between shots. (00:48:15)
Continuity mistake: When we first see Colonel Dietrich, he, Belloq and Gobler are crossing Tanis on foot. The scene cuts out when the three of them arrive next to some wagons. However, seven minutes later (in movie time) they are shown again in exactly the same position as they were in the first shot, continuing their actions. This is also seen in the two soldiers in the background on a hill and in a soldier on horseback going from left to right. Even Indy’s team is visible in the first shot, when he is supposed to be in another place only with Sallah. (00:50:09 - 00:57:00)
Continuity mistake: When Indy is climbing into the map room, he throws the staff of Ra down the hole, yet when he climbs down, it's resting perfectly against the wall, as if someone had placed it there. (00:50:40)

Continuity mistake: When Indy is being lowered into the map room, the exterior shot of the opening does not match the interior. From the outside, the wall around the opening is a lot thicker. The angle is also different. (00:50:55)
Continuity mistake: When Sallah lowers Indy into the Map Room, from the outside Sallah is standing near the bottom of the hole, but from the inside the rope hangs from near the top. (00:50:55)
Continuity mistake: When Indy is in the map room, he blows sand away from a hole and a bunch of hieroglyphics, but in the next shot, there's sand scattered all over the stone he just finished clearing away. (00:52:10)

Continuity mistake: When Indy goes to save Marion in the tent he removes the handkerchief gag. But when he decides to leave her there, he puts the gag back on and some of her hair gets caught underneath the gag on one close up shot from the left. But when the camera goes to the back of Indy, her hair is no longer caught underneath the gag. (00:56:00)

Continuity mistake: Sallah and Indy are peering into the well of souls. Sallah is to the right of Indy. Indy then drops the torch into the well, and when the camera comes back up, Sallah is on the left of Indy. (01:00:00 - 01:00:45)
Suggested correction: Indy and Sallah spent several seconds off-camera, enough to move around the hole and switch places.
Continuity mistake: When Toht frees Marion's arm, Dietrich enters the tent, and a few seconds later, Gobler enters. In the next shot, Gobler enters first, with Dietrich second. (01:07:52)
Continuity mistake: In the Well of Souls, Indy and Sallah place the Ark in the box. Indy closes the box with a wooden lid and looks up. In the next shot, a torch has appeared in Indy's hand, a rope has appeared, tied to the four sides of the box, and Sallah has appeared next to Indy's left shoulder. In addition, the box is already several metres above Indy and Sallah in the second shot. (01:09:43)
Suggested correction: There is an obvious time jump between the shots. They simply decided not to spend ten minutes showing all the details of sealing the box and tying the rope around it.
I wouldn't say this was a deliberate time jump, as they show Indy carefully and slowly placing the Ark in the box, then sealing it. In addition, when he looks up, he immediately says, "All right... take it up! Easy!" so it's clear this is a continuity mistake.
He says, "All right. Take it up" after the cut. Yes, he does look up before the cut, but since everything else is different, we are supposed to interpret it as a jump in time.
In fact, almost nothing changes between these two scenes, except for these small but clear details.
There's a deliberate jump in time between these two shots, and this transition is done creatively. First, we're below with Indiana as he places the lid on the crate and looks up toward the top entrance; it then cuts to the entrance POV, and now we're looking down as the tied-up crate is being lifted. The jump in time between these two shots is transitioned using a type of match cut. Absolutely no mistakes regarding these two consecutive shots.

Continuity mistake: When Belloq is talking to Indy and Marion in the Well of Souls, he's shown holding his hat with his right hand. But in the only over-the-shoulder shot (looking into the Well of Souls) the hat is laying on the ground and Belloq tosses a stone down with his right hand. Then the hat is back. (01:10:35)

Continuity mistake: Dietrich and Gobler walk forward to stand next to Belloq while he's saying, "Once again, Jones..." In the next shot, Gobler has disappeared (this can be spotted by the distance between Belloq and Gobler's foot in the first shot). (01:10:57)
Continuity mistake: After Indy rips a big strip of Marion's dress in the Well of Souls, it continues to change until she finally gets rid of it on the boat. At some points (eg. running from the plane before it explodes, and other shots) it seems quite straight across, more like a shorter dress than a torn one. A few moments later, when they are crouching behind an earth bank and Sallah tells Indy the Germans are putting the Ark on to a truck, the dress is badly ripped in various places across her back. Not long afterwards, it is in one piece again but back to the bottom edge being ripped. (01:13:00 - 01:21:35)
Continuity mistake: When Indiana is climbing the statue to escape from the Well of Souls, the snake he encounters makes a rattling sound. However, the snake has no rattle and rattlesnakes are indigenous to the Americas. (01:13:40)