Factual error: When the returned soldiers are on the train at the end of the film, it's an open plan post-war British Rail Mark I type, which where built from around 1950. Also the blue upholstery on the seats looks to be the corporate blue introduced by British Rail in the 1960s, used by the preserved railway owning the stock, and not what would have featured in Southern Railway carriages of the time. The carriages also have horizontally-sliding windows, which are far more contemporary than wartime trains, which had windows with a much larger vertical opening, held in place by a leather strap.
Deliberate mistake: Christopher Nolan admits he used a French destroyer instead of a British destroyer, for practical purposes (most people won't notice). The giveaway is that it has a closed bridge. British warships were built with open bridges till after the end of the Second World War, as can be seen by looking at HMS Cavalier (http://thedockyard.co.uk/explore/three-historic-warships/hms-cavalier/), which was built in 1944.
Other mistake: In the scene at the end where Farrier gazes upon his burning Spitfire on the beach, the propeller appears to be supported at the end of a simple rod. In fact the propeller would have been attached to a rather solid engine. There's also no internal structure - the entire spitfire was reduced to ashes which was impossible as this was a metal aircraft. Unlike the Wellington, Mosquito or Hurricane which were partly timber and canvas.
Factual error: When the boat returns to England they state the cliffs are Dorset. There were 3 evacuation routes from Dunkirk - all to Kent. It would be nonsense to sail from Dunkirk to Dorset as you have to almost pass Dover on the way! (never mind the fact the boat would probably not have made it without refuelling).
Other mistake: When the pilot of the Spitfire is shown ditching into the water, his engine is windmilling at high RPM as he impacts the water. This would have resulted in all of the prop blade tips being bent backwards; however, as it shows him trying to escape the sinking airplane, the prop blades are perfectly straight.
Factual error: Not until the very end of the 10-day evacuation were the French (or British for that matter) defending the very beaches themselves, and yet the movie opens with the French roadblock doing just that.
Character mistake: The story arc set on the Mole covers one week. The Highlanders attempt to refloat the beached trawler on the last day of that week. One of them confidently states that the tides are three-hourly. (The audience know this isn't true because we've seen a similar conversation between the Admiral and the Colonel.) At least some - if not all - of those Tommies have been on the beach all week. Have none of them have noticed that the tides are six-hourly?
Continuity mistake: There's a repeated issue with weather continuity throughout the movie. Many scenes cut between daylight, sunset and cloudy weather literally seconds apart. This has nothing to do with the particular time structure of the movie, since it happens between shots and reverse shots of the same scenes. These inconsistencies are particularly obvious at the opening scene, at the Stuka bombing and at the rescue of the shivering soldier. In these scenes, weather turns from heavily cloudy to fully sunny just from one shot to the next.
Factual error: Colonel Winnant wears a regimental cap badge. In fact, full colonels (as he is) and brigadiers have a different cap badge (a lion standing on a crown). It's not entirely beyond the realms of possibility that he would choose to continue to wear his old cap badge, but it would be very unusual (and completely against regulations).
Continuity mistake: As the lone Spitfire trails and attacks the German bomber, he hits the right engine, which starts smoking but the trail quickly fades. He then hits the left, which smokes a lot. We then see the same attack from below, and from this angle the both engines are smoking equally badly.