When David fears for his career after seeing the ruined painting, he incorrectly claims that he left Bean alone with the picture. It was in fact Grierson who advised him to stay. David only agreed with him; he didn't actually make the suggestion.
The painting around which the film revolves is referred to universally as "Arrangement in Grey and Black, Portrait Of The Artist's Mother" by art galleries (the first part being Whistler's original title for the painting), not "Portrait of Whistler's Mother".
When Mr. Bean unintentionally attracts the attention of two police officers at the airport they start to walk over to him. When they reach him one of the officers is different. The two officers then start to chase him and the original officer is back again.
From his side of the one-way window in the police station, Bean's tie appears straight and tidy with only the 'large' end on show. However from the police perspective on the other side of the window his tie is lop-sided with both ends clearly visible across his shirt.
Several spoken references are made to the "Royal National Gallery" as well as the labeling of an exterior shot. It should have been referred to as the "National Gallery" since the gallery was founded by the state, not by royalty.
When Bean tries to dry his trousers using the metal fan during their first meeting at the gallery, Grierson asks him to join their conversation. After Bean turns around, Peter MacNicol is clearly trying not to laugh and almost bursts out right before the scene ends.
Near the start of the film, Bean is late for his meeting. In the meeting the boss greets everyone with a "good afternoon" when only just earlier it has shown Bean look at his watch and it was around 9:00 am.
When Bean takes off on the skateboard to "fix" the painting, the camera clearly shows his feet, and he's wearing black leather shoes. Later, when he's at the security guard station at the museum, there is a shot directly of his feet, and he is shown wearing mismatched slippers. It can be explained, as part of a joke, he would wear them so he wouldn't be heard; sneaking around the gallery.