Continuity mistake: Just as the ghosts fly out of the Ark, a soldier can be seen on the left side of the altar, with nothing in his hands. Two shots later, that soldier is now carrying a banner.

Continuity mistake: As the ghosts dance around the pole to which Indy and Marion are tied, large clouds can be seen in the sky behind them. However, after a ghost circles Dietrich and passes in front of the camera, it can be seen in the background that the clouds behind Indy and Marion have disappeared.

Continuity mistake: As the ghosts dance around Dietrich, Belloq, and Toht, the Frenchman slowly looks up as the column of mist ascends upwards from the Ark, almost looking straight ahead. In the next shot, Belloq is looking down again; the shape of the mist has changed, and the column of mist is now moving to the right.

Continuity mistake: As the truck pulls up next to the Command Tent, four soldiers lift the Ark's box and a fifth soldier, positioned between the two wooden posts of the Ark's box and to Gobler's right, picks up the sticks that Indy and Sallah had used. In the next shot, that soldier is now to the left of Gobler and the Ark's box. (01:21:40)

Continuity mistake: The distance between Otto and the Nepalese changes between shots, just as Toht begins saying "Surely he told you..."
Continuity mistake: When Indy sits down in the seat of the plane look behind him, there is nobody holding a Life magazine or fitting the description of the man later shown watching him from the other seat.
Suggested correction: This is not true. Dennis Muren enters just before Indy and sits down. The camera focuses on Indy, giving Muren enough time to grab the magazine and cover his face.
Continuity mistake: In front of the idol, Indy shows Satipo the trap on the floor. Satipo's head position is different in all 3 shots in barely 2 seconds: First he's looking downwards, then straight, then his head is turned sideways, and then back to straight. (00:08:20)

Continuity mistake: The amount of moss behind the idol increases between shots.
Continuity mistake: When the portal opens over Geheimhaven, you can see that in the sky and inside the portal there are a lot of blue clouds. Two shots later, in the shot inside the portal when the fire turns on itself, the clouds have disappeared and the portal's interior is now the normal night sky (black and with one star visible).
Continuity mistake: When the Ark is 'sucking' all the fire, the klieg on the left of the screen points towards the altar. Two shots later (and after the Ark closes), that klieg points towards Indy and Marion.
Continuity mistake: Arnold holds a hot stick of metal to Marion's face in her bar, threatening her. In the opposite angle, the metal is nowhere to be seen when it should be right next to her face. Cut back to the other angle and we see it again.
Suggested correction: This entry is incorrect; the poker is always on screen, it never disappears.
Continuity mistake: In the marketplace standoff between Indy and his would-be attacker, in shots from behind Indy, he's facing one way, but in frontwards shots, he's facing a different way and when it cuts back to behind him, he's facing the other way again.
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.