Plot hole: During the climatic battle in the snow, Lyra Belacqua is about to get shot by one of the soldiers, and Iorek Byrnison comes to her rescue and is able to surprise & kill the soldier. The problems with this is where Iorek appears would have been right in the line of sight of the solider, and bearing in mind that Iorek is VERY big, wearing armour so as not to blend in with the white surroundings, and making lots of noise, it would have been impossible for Iorek to stealthily appear from nowhere.
Continuity mistake: When Lee Scoresby is talking to Iorek on Lee's balloon, Scoresby runs his fingers through his hair to pull it out of his face before he puts his hat back on. After replacing his hat atop his head, he brushes a few stray hairs behind his left ear. The shot briefly cuts to Iorek, then back to Lee. More hair than he originally had to tuck back is covering his left temple.