Factual error: At the end of the scene where CC Baxter meets the doctor for the first time in the hall, Baxter is standing in the doorway of his apartment as the doctor enters his own apartment. From that view, you can see into Baxter's kitchen, which has a window - right where the apartment the doctor just entered is located. That could not be an outside wall.
Factual error: People show up to work on 24 December (Christmas Eve), on the 26th, and on the 27th, but this would make sense only if the 24th was a Monday or a Tuesday. However in 1959 (as narrated by Jack Lemmon at the start of the movie) Christmas Eve fell on a Thursday, making the 26th/27th the weekend. Also, early in the movie C.C. Baxter says to a colleague at the office that "Mr. Heidelberg had a Halloween party last night", implying that it is November 1st, but in 1959, November 1st fell on a Sunday and Baxter would not have been at the office.
Continuity mistake: It was not a suicide attempt after all, and much relieved, CC grabs the bag of groceries to bring it inside the kitchen. He is carrying the bag to his right with the torso tilted, but in the reverse shot here he is, tilted in the opposite direction, the bag weighing against his left side. (01:37:45)
Deliberate mistake: In the bar scene with the woman blowing straw sleeves at him, Jack Lemmon and the extra next to him have their glasses full before the music ends, but empty when she walks up to him placing the rum glass on the counter. The barman brings instant off-screen refills to all of them without being reasonably enough time for him to pour. (00:52:30)
Continuity mistake: During the second round of drinks, the amount of ice left in the frozen daiquiri changes depending on the camera angle. It's poofy and opaque in the side view, and deflated and almost fully transparent in the tighter close-up on Shirley MacLaine, with an intermediate angle and state. (00:38:10)
Deliberate mistake: Jack Lemmon looks at his watch as he puts his thermometer in his mouth, and when he takes it out. Obviously in 25 seconds a 1960 mercury thermometer would not give a reading of his 101 fever, it's deliberately done for the scene to move on (although it could have been done differently, with a cut or having him retrieve the thermometer later, since the scene is static and lasts much longer, enough for a real thermometer to work). (00:19:20)
Continuity mistake: In the first elevator scene, when Shirley MacLaine calls the floor number (19), between her, Jack Lemmon and the wall there are a man with a fedora reading the New York Times, and a woman with a turban. When the elevator doors open and people come out of the cabin, the woman is nowhere to be seen. (00:17:05)
Continuity mistake: CC opens the door to Mr. Kirkeby who is back for the galoshes; you can see Jack Lemmon casting a pronounced shadow over the light switch by the door in the close-ups, but in the wider shots of the room Kirkeby is searching and then exits, that shadow is not showing, just one on the door itself. (00:06:45 - 00:07:30)
Visible crew/equipment: Randomly, a big moving shadow is cast on the underside of the flight of stairs between Daxter's apartment and the (unseen) second floor when he is unlocking the door of his flat after meeting his landlady that warned him about gas. It does not match Jack Lemmon's motions. (01:37:10)
Factual error: Jack Lemmon at the beginning of the movie lights up the pilot flame of his stove, puts a frozen dinner tray straight from the cardboard box in the refrigerator, and it's all nice and ready for him to eat (the aluminum foil is so hot it burns his fingertips, even!) in the minute and a half it took him to put the bottles out of the door and talk to his creepy doc neighbour (no cuts are shown to imply passage of time). (00:08:00 - 00:09:30)
Character mistake: The protagonist at the beginning of the movie is cleaning his own apartment when Kirkeby returns for the missing galoshes of his lady companion. Kirkeby searches a chair, looking also down on the floor to the side of it, turns around and looks up the other one and then back towards the starting point. He picks the galoshes a few seconds later finding them in a spot where he supposedly has looked already almost all the time (the actor just pretended to look). (00:06:40)