When President Mitchell leaves the helicopter, the rotors are standing still, but in a subsequent shot they are still rotating.
President Mitchell suffers a serious stroke in the fall (judging by the color of the trees). Five months later, the season appears to be the same.
In the White House kitchen, Dave asks Duane whether he carries a gun. It was the sight of Duane's gun that gave Dave a start when they first met in Dave's living room.
Correction: Duane's jacket is completely closed. You can't see his gun. Dave is rightfully startled by two people being in his living room.
When Dave pretends to have the stroke as the President in the Joint Session of Congress and is wheeled to the ambulance, it is clearly daytime. Minutes later, as the ambulance arrives at the hospital it is dusk. Seconds later, as Dave leaves the ambulance and walks away, it is clearly evening and dark. This sequence of time and lighting is clearly impossible.
Ellen is holding a grocery bag on her lap shortly after she and Dave sneak out of the White House - before they've gone to the store to buy the supplies for the sandwiches.
In the opening scene when the president's helicopter lands and President Mitchell disembarks, the engine is running and the rotors are spinning. The president is never allowed to enter or leave the aircraft while the rotors are in motion.
Dave frees-up money for a homeless shelter by guilt-tripping Cabinet members into cancelling "silly" or wasteful programs. The Federal budget is a law, passed by Congress--money allocated for one program cannot legally be spent for something else, unless the relevant law has been amended by Congress.
Togo doesn't have a king.
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 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 White House tour guide tells her audience that 42 presidents "and their wives lived in the White House" before the current president, President Mitchell. When Vice President Nance is sworn in, the TV commentator proclaims Nance will be the 45th president (as Mitchell is the 44th.) That math doesn't seem to add up. So where's the missing president? It's George Washington, who did not live in the White House, since it was completed after his presidency.
Duane leaves a car for Dave and The First Lady to go out on their own. Secret Service protocol does not allow the president to go out into public unattended, however Dave is not the president and he would have asked for the car before Ellen decided to go with him.
When Dave and First Lady are pulled over by the police and asked to step out of the car. Dave pulls his eyeglasses out of his pocket and puts them on. When he does, there is a large piece of lint hanging from the ride side of the frames. In the next shot the lint is gone.
When Bob Alexander goes public about Mitchell's involvement in the Savings & Loan fraud, he seems to forget Dave's threat in the Oval Office that he could go public about Alexander's subverting of the 25th amendment to eventually install himself as president, a much more serious crime.
The color of the Ford Tempo that Dave and Ellen drive in changes from white to light gold and back again.
When Ellen is taken to see Bill Mitchell where he is under medical care the movement of the respirator does not match the sound it is making.
Near the end of the film, when an aerial view of Washington is shown, the shadow of the helicopter doing the filming can be seen moving across the landscape.
The chamber where Dave delivers his message to Congress is obviously not the actual House chamber, which has benches rather than chairs. (It's actually the Virginia House of Delegates.)
Although set entirely in and around the District of Columbia, a Los Angeles RTD bus stop sign clearly appears in the background when the presidential limousine stops to let Dave and the First Lady out at the throng of noisy reporters.
When Dave and the First Lady leave the White House and go for a drive, they drive back up to the White House gate. A Uniformed Secret Service agent sees them, opens the gate, and lets them drive right in. There's no way a Secret Service agent would allow that without verifying they are actually the President and First Lady first.
At the end of the film when former First Lady Ellen Mitchell walks into Dave's campaign office, she is alone. First Ladies are afforded secret service protection after they leave office and would never be allowed to walk around the streets unaccompanied without an agent present.