Continuity mistake: Azog kills Fili with his arm sword, running him through from back to front. In the next moment, when Azog tosses the dying Fili off the ledge, we see the arm sword - and it is shiny, clean and spotless.
Continuity mistake: When the dwarves finally decide to leave Erebor and join the battle, they swing a giant gold pendulum into the blockade they all made earlier in the film, knocking it over to create a causeway, across which they all then run in a battle formation, straight into battle. However, just as they all exit the keep, the pendulum is on its downward swing back outside of the wall. In all following shots (from numerous different camera angles) the gold pendulum simply disappears. This is most notable in the shot taken directly in front of Thorin as the dwarves run towards the camera. The pendulum should be visible in the large hole in the wall behind them, but instead the hole is empty, with no pendulum to be seen.
Continuity mistake: During the battle, when Legolas throws Thorin's sword into the troll above Thorin, he pulls the sword out and it is stained with blood. But in a later shot, the sword cleans itself.