In the scene where Anna first meets the King, the French ambassadors are delivering "a present from the King of France" (in French in the dialogue). In the 1860's France did not have a king any longer: its leader Napoleon III had the title of "Emperor".
When Anna and King Mongkut first dance, the song being heard has full instrumentation (horns, woodwinds, etc.). But all the views of the musicians entering the ballroom and playing show only string instruments.
The King's favorite daughter, Princess Fa-Ying (Melissa Campbell), dies of cholera. Yet she appears after her death with the rest of the royal family briefly aboard the royal steamboat as the King attempts to take the royal family to safety from the insurgents marching toward the palace. Then, after the ship lands, the King and family pray at a shrine to Buddha, and Princess Fa-Ying is again seen with the rest of the children as the insurgents approach from the other side of the river.
After the fight between Prince Chulalongkorn and Louis, King Mongkut goes and investigates the reason for the fight; he sees Louis writing lines and his son just standing there. Chulalongkorn is on Anna's right while Louis is on her left. Later, when Louis is finished writing his lines he is on her right and the prince is on her left.
The Thai language used between the King and his prime minister is not royal Thai language, but everyday Thai language. Everyday Thai language is never used to speak to a member of the royal family, especially the King.
Upon introducing his queens and concubines to Anna, Mongkut states that his concubines are not as numerous as those of the Emperor of China. The Chinese emperor in 1862, the Tongzhi Emperor, was only six years old at the time and unmarried. Even after marriage, he had 5 consorts- significantly less than Mongkut's 23 wives and 42 concubines.
The goof item below may give away important plot points.
In the close-up of Prince Chowfa just before his execution, a soldier is visible behind him in the line of fire. In the opposing shot when General Alak fires, that soldier is standing aside and further away.
When the King observes his guards spotting Alak's army, the telescope is aimed to the left of the white tent behind it, and while the King looks through the telescope he aims it further to the left. In the next shot when Anna observes the King, the telescope is suddenly aimed far to the right.