The map of Germany and the Low Countries that Stokes and his men look at shows a map with modern day coastlines, including the Dutch province of Flevoland. However, the 1,000 square kilometer insular province only came into being when it was reclaimed from the sea in 1968, well after the Second World War. Also, the "Afsluitdijk" (a 32km long dam sealing off the Zuiderzee) is not shown on the map, despite having been completed in 1933.
During live fire training, the location of the World War I memorial is attributed to Saint Louis, Missouri. The sculptor upon whom John Goodman's character is based designed sculptures for St. Louis' World War I memorial, which is called the Soldiers Memorial and should not be confused with the National World War I Memorial located in Kansas City, Missouri.
One of the Monuments Men is taken for a ride on a biplane that has been hidden in a French barn. The aircraft registration number starts with a G, indicating that it was not registered in France, but in Great Britain.
When US Army Lieutenant James Granger visits the apartment of Claire Simon, he is not wearing his uniform but is wearing a casual, short-sleeve shirt instead. During World War II, it was a strict rule that members of the military had to wear their uniforms at all times, on and off base, even while home on leave. Liberated Paris in 1944 was still considered a war zone, making his disregard for uniform especially egregious.
The first mention of the Madonna is in connection with the Onze-Lieve-Vrouwekerk ("Church of Our Lady"). The church shown in the film, though, is a completely different one. Also, the Madonna is displayed in an altar piece, not standing on its own.
Prior to the scene where Generals Patton, Bradley and Eisenhower are seen in a "photo-op" involving recovered treasure, a row of parked jeeps is shown with stars on red bumper plates belonging to the generals. One jeep (presumably Eisenhower's) has five stars in a horizontal row, whereas the actual rank insignia and bumper plate would have the five stars arranged in a ring with an empty pentagon-shape in the center. Also, that jeep and the one beside it with four stars (presumably Patton's) both have bumper numbers that begin with "3A" indicating both vehicles are assigned to Patton's Third Army (as are most of the vehicles used by the Monuments Men themselves); as Supreme Commander of the European Theater of Operations, it's highly unlikely that Eisenhower would have had to borrow one of Patton's jeeps.
The goof items below may give away important plot points.
When old Stokes comes back to Bruges to see the Madonna in 1977, he can come close to the statue to view it. The statue has been behind bulletproof glass since 1972 and the public can not come closer than 15 feet.