Continuity mistake: When Ted visits the Once-ler for the second time and dives to the ground to avoid being launched by the sledgehammer as happened the first time, he stands up and his goggles are around his neck but hanging behind him. The scene then switches to the Once-ler and then back to Ted, and his goggles are still around his neck but now hanging in front of him.
Other mistake: In the opening song, at the end when the people are "thhhhhhhhhneeding" it shows a closeup of a little girl with red hair and a yellow bow, you later see her in the flashbacks multiple times. She is throwing tomatoes at the Once-ler and singing along to the Thneed song. There are also 2 O-Hare workers, most noticeably during the opening song. They sing the part about the tree needing 96 batteries. You later see them in a flash back during the "Everybody Needs a Thneed" song. You see them well, in the front row near the end of the song, O-Hare uniforms and all.
Continuity mistake: When Ted is playing a board game with his family, his grandmother's letters in her tray change without her touching them. We first see them from her perspective as "LORXBR-". The last one is cut off, but you can see that there is a letter ball there. Two shots later, the O is missing. She now has "L_RXBRA". The grandmother and the letters are visible in each shot, so she cannot have picked it up without us seeing.