Continuity mistake: When Oskar shows his Rubik's Cube to Eli and explains how it works, one side is all white except one square, which differs from being red vs. blue between shots.
Other mistake: When Oskar finds the solved Rubik's Cube it cuts to a shot of Eli sleeping, wearing a large shirt/dress. However, this shirt/dress is the same one that she takes from Oskar's mother's closet later on in the movie and therefore, she wouldn't have it yet.
Plot hole: After Eli's guardian is hospitalised for acid-burning his face, Eli hears on the radio that they can't ID him because of his facial wounds. When we see him in the hospital however, only half his face is injured, the other half is fine which would make a mirror composite easy for lab techs.
Continuity mistake: When Oskar intercepts Lacke's plan to kill Eli in the bathtub, Eli kills Lacke and drinks some of his blood, leaving some blood on her lips. She then kisses Oskar, which in turn leaves some blood on his lips. In the cut right after that the blood is gone from his lips. Later on it appears again, when he tastes it with his tongue.
Continuity mistake: When Eli runs out of the basement and then runs outside, after she scares Oskar by licking his blood off the floor, there is snow everywhere. She then climbs up a tree, and when she is at the top of the tree there is no snow anywhere to be seen in the background. From the following day on the snow is back.
Continuity mistake: One of the bullying scenes has Oskar getting hit across the face with a branch, which leaves a long red cut. Shortly after that he goes to visit his father, and during this time both the bandage as well as the scar itself are gone. Later on in the movie they both reappear.
Visible crew/equipment: When the bullies first enter the swimming pool area to punish Oskar, one can see the film crew's reflection in the glass at the other end of the pool area.
Factual error: When Virginia is attacked by Gosta's cats and throws herself down the flight of stairs to get them off her, her clothes aren't ripped or messed up afterward the way they should have been after such a vicious attack.