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.
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 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.
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.)
Lord Wendover is introduced to Barry as "Gustavus Aldophus, Earl of Wendover". However, later on when Barry (now in disgrace) attempts to persuade the Earl to join him for lunch in the Club dining room, he addresses him twice as "Neville".
When Lord Bullington interrupts the music recital to announce his leaving home, his hairstyle goes from fluffy and covering his forehead to being combed hideously upward while verbally blasting his stepfather, then returning back to it's previous fluffy state after his mother goes crying off, just before an enraged Barry springs from his chair and starts beating him.
When Capt. Potsdorf first encounters the so-called Lt. Fakenham, in most instances the former refers to the latter using the British pronunciation ("lef-ten'int"); but on at least one occasion he uses the American pronunciation ("loo-ten'int").
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).
When Barry and Lord Bullingdon fight a duel, a coin is tossed to determine who shoots first. When alternate shots are taken in a pistol duel, the challenged always shoots first and the challenger shoots second. Barry should have shot first followed by Lord Bullingdon.
The narrator states, early on, "About this time, the United Kingdom was in a state of great excitement". The United Kingdom came into being in 1801, when it merged with the Kingdom of Ireland, before which it was known merely as the Kingdom of Great Britain.
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 it wanted to reconquer with its allies. Meanwhile, Britain and France had been 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.
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 #39 and the dark flag looks like it could be the flag of Infantry Regiments #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 #13, which was known as a hard-fighting, hard-drinking unit.
When Lord Bullingdon is returning back to challenge Barry for duel, he is walking slowly in the corridors of the mansion full with drunk people. On the table on the right is visible Dom Perignon champagne bottle. Actually in 17th century Dom Perignon was a monk who experimented with sparkling wines, but the bottle and the label appeared at the end of 19th century.
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.
In the 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 useless tool for cutting wood--chopping and splitting, yes, but not cutting.
The goof items below may give away important plot points.
The narrator reads from the Saint James' Chronicle: "Died at Spa in the kingdom of Belgium: the right honorable Sir Charles Reginald Lyndon". That kingdom was created in 1830. In 1772, most of future Belgium was a collection of counties known collectively as the Austrian Netherlands, but Spa was in the autonomous Prince-Bishopric of Liège.