Continuity mistake: At the beginning of the movie, in the shot after Roscoe says "He's got our thing!", the lamp behind him and the lamp in front of Fedora disappear. Also, just after Roscoe says that line, immediately his voice is heard saying "Get him!", but his lips don't move. (00:04:44)
Other mistake: After Young Indy runs out of the cave with the Cross of Coronado, when he whistles for his horse and jumps down, note the horse's facial marking and leg marking. When Indy mounts the horse it's a different horse, note its markings, and as he rides toward the circus train the horse has changed again. (00:04:55)
Visible crew/equipment: After Young Indy locks the two treasure hunters in the reptile train car he pulls a snake out of his shirt, and just as the third young treasure hunter tackles Indy we can see a crewmember's arm and equipment at the left side of the screen, before it cuts to the long shot. (00:07:00)
Continuity mistake: In the beginning where Indy is young, he fights some guys on that zoo train. When he's on the crate with the rhino in it, the background changes in each shot. Once there are trees behind and a split of a second later there are rocks, once there's a cliff and once a plain, and it all changes in splits of seconds as they change shots. (00:07:25)
Audio problem: When Indiana is fighting one of the bad guys on top of the rhino's car on the train, the rhino sticks his horn straight between Indy's legs. Indy says, "Holy smokes" but if you look, he actually says something else. (00:07:35)
Continuity mistake: When young Indy (River Phoenix) is advised by the fedora-topped henchman at the end of the chase, the cut he received while using the bullwhip in a previous scene changes angles. This is a flipped shot, note Indy's backwards shirt. (00:08:25 - 00:11:45)
Visible crew/equipment: When Young Indy is trapped in the train car with the lion, he tosses up the whip to Fedora, and as he's being pulled up in the interior shot, he's wearing a stunt apparatus around his waist above his beaded belt. [The stunt double is also wearing a glove on his right hand holding what looks like skewered meat, to presumably draw the trained lion, but that's a lot harder to see without pausing]. (00:08:35)
Continuity mistake: In the beginning, when Indy is fighting aboard the Coronado ship, the very first time we see Indy in this scene (before we see him get punched), he is bleeding from the right side of his mouth. Throughout the remainder of the scene, he is bleeding from the left side of his mouth. (00:11:50)
Continuity mistake: During the fight on the Coronado's deck, Panama Hat is climbing the ladder with his back to the camera. In the next shot, when he says, 'Grab him, he's getting away!', he faces the camera. (00:13:21)
Continuity mistake: The scene when the barrel falls on the TNT box and explodes is flipped (Panama Hat is on the opposite stairs). (00:13:44)
Factual error: At the end of the classroom scene, Indy gives Marcus the Cross of Coronado. There is a pile of books on Indy's desk that he laters picks up and takes with him. One of the books (with the whitish spine, under the red book) is "Living Egypt", written by Paul Strand and James Aldridge. This book wasn't published until 1969, over 30 years after this movie takes place (1938). (00:15:20)
Factual error: As Indiana Jones opens the door to Dr. Jones' house, note the light switches just inside the front door. They are the kind developed in the late 1970s and still in use today, not consistent with the film's setting of 1938. (00:22:30)
Continuity mistake: When Indy takes the mail out of his pocket in the beginning of the film, he keeps the "Grail Diary" in his hand, but puts the letters down twice. Once before he opens the package and once after with the wrapper. (00:23:00)
Factual error: The stamps on the mail at Sr. Jones house are Texas Statehood issued in 1945. A bit late for the scene. (00:24:15)
Factual error: Indiana Jones checks his dad's grail diary to find the picture of the stained glass window in the church in Venice. With the book open to that page, the picture is on the left-side page, and under Indy's thumb on the right-side page is a snippet of an article about marriage statistics. The article mentions statistics up to and including the year 1943. Problem is, the movie takes place in 1938. (00:27:00)
Continuity mistake: In the library scene Indy discovers the "X" high up on the balcony. The X is green with a grey background. When he breaks the tile to find the tomb the X has become a faint outline on the floor. (00:27:40 - 00:28:45)
Suggested correction: You still can see one "leg" of the X on the floor, it's only darker than viewed from above because the camera angle and illumination set used.
I think it is meant to be an optical illusion.
The "X" is first shown as a dark green "X" on a beige background. Next, we are shown the same dark green "X" that is barely visible over a green background. I think we are meant to understand that the beige square tiles were lifted away in a cut scene.
I see no reason why they would replace the floor just for the higher shot, it's the same floor throughout the scene. When they enter it's the same floor we see later as they are going into the hole. It's probably not a real marble floor, so they can use a styrofoam or plywood tile that Harrison can lift, one that matches the surrounding tiles. They don't shine as much as the rest of the floor. In the shot up high there is different lighting, so that could explain it. It just appears to be different. Of course, sudden different light can be seen as a revealing mistake.
Suggested correction: Not a mistake, just a different viewing angle.
Revealing mistake: When Indy has just climbed the stairs in the library and is leaning over the rail to look at the X below, directly behind him is one of the worst props I've ever seen - a flat fake book case. (00:29:00)
Visible crew/equipment: In the catacombs after Indy passes the picture of the Ark Of The Covenant and approaches the next wall with the lighter, (when the screen goes upwards a bit) the shadow of a boom mic can be seen dropping into view at the top of the lighted wall. (00:31:40)
Continuity mistake: In the catacombs underneath the library, when he find the "X" on the wall, he is shown ramming the wall twice to break through. The first time he rams it, however, the wall looks as if he had already had a few goes already. (00:31:55)
Revealing mistake: When Indy is breaking through the wall in the catacombs, and he uses his shoulder as a battering ram. You can see his head actually bounce off a piece of the rock real hard as he is going through the wall. If the rock was real, I guess Indy would have a lot to cry about that night. (00:32:00)
Suggested correction: This rock is real but very old. That's why it was easy to break.