Derek Luke (2nd Staff Sergeant Aubrey Stamps) continually uses the slang term "book" meaning, to leave abruptly, which was not used until the late 1960s, nearly a quarter of a century after the time in which the film was set.
One of the soldiers was playing with Italian kids saying Up High, Down Low, Too Slow. According to Wikipedia, high fives started in the 60s. It also says, "The 'low five' had already been known, during the 1940s, in African-American Vernacular English (AAVE) as 'giving skin' or 'slapping skin'." This scene wouldn't have happened in the 1940s.
Many of the German soldiers carry K98k rifles that are missing sight hoods and cleaning rods. These parts being missing indicate that the weapons were likely captured and stripped down by the Soviets during or after the war. A rifle in German service would have these parts.
When the guys leave the town in Louisiana there is a scene where they turn around on the railroad crossing. The scene and the movie is set in the 1940s. Concrete rail crossings did not start making an appearance until the 2000s.
When the professor examines the marble head he recognizes it as the "Primavera" from Santa Trinità in Florence. He assumes it is by Bartolomeo Ammannati, but the bridge is by Ammannati; the sculpture is by Pietro Francavilla.
The revision of the Geneva Convention in regards to civilians (4th Geneva Convention) was not adopted until 1949, therefore when the German Colonel is telling the Captain to kill 10 Italians for every German lost and the Captain refers to that violating the Geneva Convention would not have taken place. However, the Hague Convention IV could have been considered in parts, since the 4th Geneva was adopted in parts from the Hague.
Before arriving at General Almond's headquarters, the caption reads "92nd Infantry Division Regimental Headquarters."
This is clearly the Division Headquarters as General Almond is the division commander of the 92nd Infantry division and wears the two stars of a major general.
The statue head used as a plot device is described in the movie as "white marble". The density of marble is about 160 pounds per cubic foot. Throughout the movie it dangles off the pack of the actor, bouncing off his leg as he walks or runs. Also, the actors cradle it in their lap or on their chest made it clear the object was definitely not made of stone.
Just before the scene in which the aerial photo of German troop buildup is presented to the Division Commander, the flag over the entrance to the building is hung incorrectly. The field of stars is in the upper right hand corner whereas it should be in the upper left hand corner when hung vertically.
The goof items below may give away important plot points.
In the last scene, the main character is standing "alone" on a deserted beach, and does not see the man in the beach chair 50 yards to his right, with the head of the sculpture (the central item in the entire movie) on a table, until another character shows him.
When Bishop is killed, his dog tags are bloodied and one even has a bullet hole through it. However, later when they are placing Hector on the stretcher and they are identifying his fallen friends by their dog tags. Bishop's dog tags are almost like new.