After Bob Alexander has his press conference implicating the President in the First Liberty scandal, the President, the First Lady, and Alan go to meet with federal regulators/lobbyists. When the First Lady steps out of the limousine, she's wearing a red dress. When everybody is inside at the meeting, she's wearing a gray dress.
When Dave is asked to call his office and explain why he will be away, he is wearing President Mitchell's glasses. In the very next shot, as he begins to make the call, the glasses are on the desk in front of him.
In a latter scene, Dave is meeting the VP in the Oval Office. As they enter the office, the tall clock in the background shows 10:30. Several minutes later the clock shows again in the background. The time is still 10:30.
The Ford Tempo that Dave and the First Lady are riding in has automatic shoulder belts when they're stopped by the police officer, but a manual seat belt on the driver's side when they return to the White House.
In the final scene in the movie when Dave and the widowed First Lady step into the room, close the door and draw the blinds, the blinds are moving. Just before Duane steps in front of the door, the blinds are still.
In a scene shortly after Dave takes over as President, Bob Alexander and Alan Reed are talking in an office. Bob gives Alan a glass of water with Alka-Seltzer in it. Alan takes a sip before setting the glass down, but when he picks up the glass again during the next shot, there is no Alka-Seltzer in it, only water.
When Dave removes a hankie from his tote bag for Ellen as she starts to cry, it is crumpled between his index and middle finger. When she reaches for it it is neatly folded between his thumb and index finger.
Dave and the First Lady take a ride at night, and the police stop them for making an illegal left hand turn. When Dave is trying to convince the police they are "look-alikes", he puts on his glasses, and there is a strange string of plastic swinging from the right corner of the lens. It hits him in the face, then in the next scene, it is gone.
When Dave is taken from the Capitol to the hospital he is taken to the Bethesda Naval Center in Bethesda, MD (based on the visual of that facility). This clearly is not the closest trauma hospital to the Capitol and would not be where the President would have been taken.
When Dave/President Mitchell, fakes his second stroke, he is taken from the Capital by motorcade to the Hospital. The driveway actually is leaving the International Trade Center/Ronald Reagon Building, onto 12th St, Heading south to the National Mall. This is 11 Blocks West of the Capital. Also the Opposite Direction from Bethesda Naval Hospital.
In the very first scene, as Marine One (the helicopter carrying President and Mrs. Mitchell) approaches the White House, a shot from the pilot's point of view shows the American flag flying above the executive mansion. The flag only flies when the president is actually in the White House.
When Dave, Bob Alexander and Alan Reed are in the press briefing room for the first time, the presidential flag and the American flag in the wrong positions. Federal law requires that the American flag to be placed to the right of any other flags (unless it is center of a group). In the shot, it is placed to the left (from the speaker's point of view) of the presidential flag.
When Dave holds the joint session of Congress, the members of Congress are sitting at desks; thus, it appears as if the session is occurring in the Senate chamber. A joint session would be held in the much larger House of Representative chamber, which doesn't have desks for every member like the Senate chamber does.
When Dave calls the Joint Session of Congress, none of the US Supreme Court Justices are present. They would be seated at the front of the chamber dressed in their robes. For none of the 9 judges to attend such an important Presidential address would be unthinkable.
The goof item below may give away important plot points.
After the speech before the joint session of Congress when Dave, as President Mitchell, admits wrongdoing in the S&L scandal and then suffers his second and final "stroke," as the ambulance is arriving at the hospital, the news report says that "the stroke hit at 8:42 pm..." The speech was clearly delivered in the daytime. Not only was it light out as everyone was filing into House chamber, but the clock on the wall of the House chamber read 1:25 pm. The "stroke" hit shortly after the beginning of the speech.