The newspaper article that Northbrook reads at the beginning of the movie states that the King of Carpathia's name is Nicholas. In the end credits, the character's name is listed as Nicolas.
However, it is relatively common (especially in the era portrayed in the film) for newspapers to make errors in spelling.
At the beginning, inside the Carpathian embassy, the two men stand at the window as the royal carriage carrying the King and Dowager passes. The carriage doesn't just move past the window, it rolls along; it and the avenue trees traverse past the open window, as if the men are standing still and the rest of the outside world is parading magically past the window opening.
Northbrook refers to the foxtrot, a dance that didn't premiere until 1914, three years later.
In the shot overlooking the Carpathian Embassy, the rear of Buckingham Palace is shown. The Palace has a small lake and a wooded area at the far end of the vast expanse of the lawn. Both are missing in the shot.
The initial titles show a pan across London that is clearly made from the south bank of the Thames. While it correctly shows the monument to the Great Fire of London to the west of the Tower of London, it then shows the Houses of Parliament way to the east of these. Parliament is way to the west of both in Westminster.
(at around 1h 49 mins) A boom shadow falls on Elsie (Marilyn Monroe) and stays there for several seconds.
When the prince's mother-in-law speaks to the showgirl (Marilyn Monroe) in French, the actress' American accent is so strong and her pronunciation so inaccurate that most of her French is completely unintelligible to Francophones, save for a word here and there. Yet, the character is Hungarian and should have had no American accent when speaking French.