Rudy Steiner wears a winter sweater with a metal zipper. Buttons on children's clothing are much cheaper and much more likely with war-time metal shortages in 1942 Germany. The zipper was mostly seen as a novelty item in this era.
When Liesel is climbing in the window to steal books from the library, she knocks two books off the table. When she picks them up, the direction the books are facing changes between shots.
When Liesel is seeing Hans off at the train depot when he's been conscripted, she is standing several cars down from the engine, facing toward the front of the train as Hans steps aboard one of the passenger cars and we can see at least one other such car between his and the engine. The camera angle then changes and we see Liesel has not moved but as the train begins to move forward, she is beside the coal car.
During the book-burning sequence, a cage-like sub-structure can be seen where the book props have burned away from their mound-shaped form.
While in the basement, Hans comments that the snowman will not melt because "it's freezing down here," yet no character's breath can be seen in the air, despite the fact that all are breathing heavily. The visibility of breath in the cold is determined by not only the temperature, but also the relative humidity, so it is possible for it to be cold without the characters' breath showing.
During the September 1939 scene, a boy on a bicycle holds a newspaper and excitedly exclaims Hitler has declared war. This is factually incorrect. Britain and France both officially declared war on Germany on 3 September 1939. Germany did not declare war on either nations. Hitler was hoping both Britain and France would come to the negotiating table as they had done previously over Czechoslovakia (The 1938 Munich Agreement). This led to a period known as "The Phoney War" when both side did little after the fall of Poland. The boy doesn't actually say that Germany declared war. His exact lines are, "England declared war on us! We're at war with England!"