Continuity mistake: When President Mitchell arrives at the White House by helicopter someone hands him the leashes of his two dogs. Mitchell winds the leashes around each other almost all the way down. In the next shot the leashes a completely separate again. (00:02:15)
Continuity mistake: When the former First Lady comes to Dave's office they start kissing. She puts her right arm first around his waist, then around his shoulder. Seen through the office door it's down at his hip again. (01:41:05)
Continuity mistake: Dave makes a submarine sandwich for Duane and himself. He is seen shredding some carrot on the still open sandwich, then talking to Duane without moving his arms, and finally there is a close-up where he cuts the closed sandwich into half. (00:40:35)
Continuity mistake: When Dave and the President's wife are going down the elevator to see the real President, right before the elevator stops she unfolds her arms and her hands are at her side. The elevator door opens and her arms are folded again. (01:01:50)
Factual error: Nance give the President fertility beads from the King of Togo. Togo is not a Kingdom - it's a Republic.
Continuity mistake: When Dave starts working as President Mitchell after his 'illness' and sits down behind the desk the chair topples over backwards. It's a funny scene but, according to even the lowest office furniture safety standards, absolutely unimaginable. (00:28:30)
Continuity mistake: When President Mitchell has a staff meeting after he arrives at the White House by helicopter, there is first one, then two empty chairs between him and Alan Reed. (00:04:50)
Continuity mistake: When Dave comes to the President's office where VP Nance is waiting, Duane is holding a pile of newspapers in his right hand. After he explains to Dave whom he is going to see he puts his left thumb under his belt. The camera cuts briefly to Mitchell's former girlfriend, and when we see Duane next he is holding the papers firmly in both hands, which is theoretically possible but looks quite wrong. (01:21:05)
Continuity mistake: At the end of Dave's first entrance as President Mitchell in the hotel lobby, Duane grabs him by the arm and pulls him away from the crowd. When the camera angle changes Dave is surrounded by other security staff but Duane isn't there. (00:14:20)
Continuity mistake: When Dave and the First Lady go to the homeless children's shelter in the President's limo her skirt slips off her knee. Her hands are folded on her left side, as she was trying to lean away from Dave, but when we see the knee from Dave's perspective her hands are folded in the lap. (00:42:50)
Continuity mistake: When Dave starts acting as President, Bob asks him to call his company to tell them that he is going on a trip. When he makes the call and the camera angle changes the glasses he was wearing are lying on the desk. (00:28:30)
Continuity mistake: When Dave gives his first speech on the balcony, the crowd on the lawn changes shape between the back and front shots.
Continuity mistake: While Dave's in the shower talking to the First Lady, his fringe changes style all the time, despite the water not running.
Continuity mistake: On their escape from the White House Dave and the First Lady get stopped by the police. When Dave starts to act as a Mitchell impersonator he puts on his glasses. There's some fluff hanging from the ear piece which is gone in the next shot. (01:08:35)
Continuity mistake: The DC police cars escorting the president after the "second" stroke to Bethesda Naval Hospital would not have happened that way. Once Ambulance and Presidential motorcade reached Maryland lines (Bethesda Naval Hospital in Maryland) outside DC, then the Maryland state police escort or local county police escort would take over. DC police would not cross into Maryland as an escort.
Chosen answer: Dave and the First Lady snuck out of the White House via a secret underground route which Dave, inexplicably, knew about but the First Lady didn't. The passage led them from inside the White House to an outside location, nearby where Duane of Dave's secret service detail had left a car for them to use. It's still implausible that they could circumvent the remaining Secret Service in making their sojourn, but that is the disbelief that we, the audience, are asked to suspend for the sake of advancing the plot. Upon returning, the couple drive up to the entry gate, where the guard, seeing whom he believes to be the President and First Lady, lets them return to the house. Their only explanation?"The President wanted some ice cream." The MORE implausible question would be how the guard simply trusts that two look-alikes driving up to the White House unattended in a non-government car with a lame excuse would simply let the two pass without consulting the secret service, who should have informed him of the President's comings and goings?
Michael Albert