When Akeem and Lisa were walking through the park at night, just after they walked under a bridge, you could seen the twin towers in the background. Then they came upon the Duke brothers sleeping in the rubble. Akeem gave them the wad of money, then he and Lisa started walking again. They immediately walked under the SAME BRIDGE again, and there you could see the twin towers in the background again. Somehow the Duke brothers moved without moving!
At the breakfast table, King Jaffe Joffer is surprised by the fact that Prince Akeem has grown a mustache, but later in the movie we see pictures of Akeem on the Zamundan 100-Pound note and on the Zamundan coin (the one he flips to decide whether to go to Los Angeles or New York) with him clearly depicted having a mustache. Unless that version of the money was minted VERY recently, then it's very unlikely for the King to not have noticed back then. Also in the picture of the King, Queen and Akeem seen in the Zamunda exhibit at the museum in New York, the picture shows Akeem with a mustache as well.
Akeem is introduced to the "Princess of Finland", who speaks Swedish. Finland is not a kingdom, but Sweden is, and it currently has two princesses. It's possible that the person making the introduction was mistaken.
When the landlord describes the apartment to Akeem, he says it only has one window facing a brick wall. When Akeem and Semi are on the fire escape there are clearly two windows, both of which face the street and a lot more than that is shown in the shots just before.
When Semmi picks up the Hold-Up Man's shotgun, he cocks it and points it at him. Since the robber fired a shot into the ceiling moments before, the shotgun should have ejected a spent shell, but it didn't.
In the burger bar, the King's Aide throws the money on the counter and Mr. McDowell picks the note up to examine it. In the next shot, the note can still be seen on the counter and as the shot ends, a hand starts to reach for it (it appears that the order of the shots was reversed).
When Oha pays Cleo "for his trouble", the bill is crumpled when it is put on the counter. When Cleo picks it up for the close up, the bill is crisp and flat. This happens again when Cleo pulls the bill from his pocket to show to Lisa.
As Lisa enters the subway station just before Akeem confronts her, she has been walking in the pouring rain and her coat is soaked. Moments later as she speaks with Akeem, her coat is almost completely dry. Akeem's clothes show no signs of wetness at all, despite running from the car to the subway station in heavy rain.
As Akeem and Lisa are walking, right before Akeem gives Randolf and Mortimer Duke the wad of cash, the twin towers of the WTC are visible and pass in the background as they walk. Akeem stops and gives the brothers the cash and he and Lisa begin walking again in the same direction as before. As they continue, the twin towers again pass in the background.
When Lisa throws the earrings back at Akeem, she throws them overhand in a downward direction, too low for Akeem to have caught them without bending down. When we see a shot of Akeem, he catches the earrings at his waist level, and the earrings are seen tossed upward as if underhand.
When Lisa gets off the subway after Akeem confronts her, the station is virtually empty. When the subway speeds away, a large group of people, most likely a crowd watching filming take place, can be seen reflected in the windows.
When the landlord shows the apartment, there are two windows side by side and a third facing the bed (you know this because there is sun light above) suggesting that part of the apartment sticks out. In the next scene, when Akeem is on the balcony, the windows are spaced about two feet apart, and the third window previously mentioned couldn't exist since the building is flush, which you can see when Akeem yells "Good morning my neighbors!"
In the scene where Cleo McDowell is telling Lisa Akeem has his own money, the bill is wrinkled as he pulls it out of his pocket, but when Lisa stares at the bill it is completely straight. When she hands it back to her father the bill is wrinkled again.
At the wedding scene at the beginning of the movie, and when the camera pans down to the throne. A women in double denim can be seen standing in the back next to a guard on the left side. This has to be a crew member since that the clothes she wearing does not match the rest of the guest, and the fact that she is standing very close to the royal family.
In a scene where Mr. McDowell is in his office, he picks up the company manual and it says "McDonald's." The restaurant is called McDowell's, a parody of McDonald's, but the manual actually says McDonald's on it. However, that is actually the joke - since McDowell's has deliberately copied every aspect of McDonald's (as evidenced by the guy taking pictures of the restaurant earlier in the film), it would not be surprising that Mr. McDowell would have a copy of the manual.
When Akeem and Semi arrive in America, Akeem has obviously never heard a certain four-letter "F" word given that he says "What does dumb fuck mean?" in the cab ride and also shouts "Fuck you, too!" gleefully to his neighbors from his apartment fire escape. Yet, when the criminal (Samuel L. Jackson) attempts to hold up McDowell's and uses the same foul language, Akeem tells him to watch his language. However, Akeem could (and most likely would) have figured out the real meaning of the "F" word while he was staying in America.
When the robber enters McDowells, he walks through the door, pulls the shotgun out from underneath his overcoat, and shoots into the ceiling. Unfortunately, the debris from the ceiling begins to fall slightly ahead of the shot being fired.
When the landlord describes the apartment to Akeem and Semi, he says "it's only got one window, facing a brick wall." In the next scene, Akeem is out on a balcony which is accessed through a window. In the same shot, Semi opens another window directly to Akeem's left to speak with him. There is also another window directly to Akeem's right which appears to be from the same apartment.