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)
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)