Theodore Groves is referred to in both dialogue and the credits as "Lieutenant Commander". However, this rank wasn't created in the British Navy until 1877, and 'On Stranger Tides' is apparently set in the 1740s. Groves can only be a Lieutenant or a Commander, with the latter being more likely in serving as second-in-command to an Privateer Admiral (Barbossa).
In the final scene between Jack Sparrow and Angelica, her hair moves around between shots. When the camera is over her left shoulder, on Jack, the left side of her neck / shoulder is visible, as well as the earring in her left ear, but from all other camera angles, her neck and the left earring are complete hidden from view by her hair.
At the end of the movie Barbossa becomes captain of the Queen Anne's Revenge. We see him ripping up a letter twice, thus the letter would be in four pieces. We see a shot of the pieces flying away in the wind, however there are many more than four.
When the Spanish gatecrash the fight at the Fountain of Youth and the British officer makes his stirring speech while holding the Union Flag (the anachronous nature of which has already been covered), he's holding the flag upside down.
Barbossa's ship, the HMS Providence, flies a version of the Union Flag (aka the Union Jack) that was not adopted until 1801. The flag used by the Kingdom of Great Britain during the reign of George II comprised the cross of St. Andrews surmounted by the cross of St. George; the cross of St. Patrick was not added until the flag was updated in 1801.
Jack and Angelica jump into the Thames to escape King George's soldiers, and are shown swimming in clear water. The River Thames of the 1740s was a slurry of waste and a fall into the water could - and often did - kill.
When Phillip takes off his shirt to give it to Syrena, a smallpox vaccination is visible on his upper right arm. Although smallpox vaccinations were not available in the 1700s, various forms of inoculation were used as early as 1720.
When Jack's hands are tied and he tries to sip from the hollow peg leg, he can't raise it very high and there is a sound like slurping. But Barbossa has already raised it much higher and taken a long drink. There should be nothing in the top inch to slurp.
King George II was German. He learned English when the family became heirs to the British throne. Most probably he spoke with a German accent, but not necessarily because education in different European languages by teachers from the respective countries was common for noble house in those days.
When Blackbeard threatens Jack with a pistol, he is heard cocking it, but when he holds it up to the camera we can see that it is only on half-cock. A pistol cannot be fired in the half-cock position, nor can it be loaded in the fired position, implying that either it is not loaded, or a technical error has been made by the film makers.
When Jack escapes from the Spaniards, he first drops coconuts from the top of a palm, knocking out one soldier who falls to the ground, near the palm. In the following shots of him incapacitating all the soldiers, there is nobody on the ground there.
When the Spaniard stomps on the silver chalices, it creates broken gaps from where the cups' walls meet their silver rims. Later, when Angelica takes a sip from a chalice, you can clearly see she drinks from a section that is broken. Any liquid in the cup would pour out of the gap before it reached her lips.