When Barry and Bullingdon fight, there is a map of the world in the background, with a decorative picture of a steam train on it. The film is set in the 1700s, but the steam train was not invented until the 1820s.
When Barry is escorted to the Prussian border, in the disguise of the Chevalier, when the coach and horses leaves the château, wide tracks (that could only be made by modern vehicles) are clearly visible on the gravel drive.
Captain Potzdorf and Colonel Bulow (the commander of the Prussian regiment into which Barry was pressed), wear mustaches. No officers in the Prussian army wore facial hair, with the exception of hussar (light cavalry) officers. Facial hair fell out of fashion for gentlemen (and therefore, officers, who were considered gentlemen), from the end of the 17th century until after the beginning of the 19th. Mustaches became common in the German army only after that time.
When Barry disguises himself as Lt. Fakenham to desert from British army, he should have riding boots. Instead, he wears the gaiters that soldiers serving on foot wore. That would stand out, to any soldiers of the day, and lead to his immediate detection. (Potzdorf, for example, wears riding boots; as a captain, his rank would require him to ride.)
While the Chevalier is gambling with Lord Ludd (after he and Barry escape into Saxony), we see the Chevalier cheating: there is a cut to his right hand, under the table, where he drops a card (the four of clubs) from his sleeve and into the palm of his hand. Immediately before this cut, his right hand is resting on the gaming table and his left arm is by his side. Immediately after this cheat-cut, his right hand remains on the table and he brings his left hand above the table to drop the palmed card (the card that we just saw him palm into his right hand).
The flags shown in the scenes of Barry's service in the Prussian army are from three different regiments. The flag with a white background and grey/silver-grey vertical stripes is that of the Royal Guard regiment, the yellow with white corner darts is that of Fusilier Regiment Nr 39, and the dark flag looks like it could be the flag of Infantry Regiments Nr 2, 8, 13, 28 or 54, all of which had black flags with varying colored details. The regiment's uniform distinctions of white lapels, cuffs and collar make it look like Regiment Nr 13, which was known as a hard-fighting, hard-drinking unit.
The Narrator says: "It would require a great philosopher and historian to explain the causes of the famous Seven Years' War." Actually, the reasons for the Seven Years' War are quite clear and well known. Prussia had invaded a rich Austrian region 16 years before, which Austria wanted to reconquer with its allies. Meanwhile, Britain and France were fighting for almost a century (with some relative peaceful years) over worldwide colonies. Stanley Kubrick probably wanted to make a statement about the uselessness of war.
In one part of the film, Barry is rowing a boat peacefully on a lake with a dog sitting obediently in the boat. The dog is a Yellow Labrador which was first bred in 1899. But though it's not yet christened as "Yellow Labrador" and not yet intentionally bred, there's no reason why a dog like this shouldn't exist in that time.
A short scene where Barry is cutting firewood with an axe. The wood in the pile and the piece he is cutting all have cleanly sawed ends. If he had a saw, he would be using it, as an axe is a poor tool for cutting wood. Chopping and splitting, yes, but not cutting.
The goof items below may give away important plot points.
When Sir Charles Reginald Lyndon dies, the voice over says that the Saint James' Chronicle wrote: "Died at Spa in the kingdom of Belgium: the right honorable Sir Charles Reginald Lyndon". Belgium didn't exist as a kingdom until 1830. In the mid-1700s it was a colony called Austrian Netherlands.