At the beginning, when Susan picks up a magazine from the news stand, it is apparently a December 1939 edition magazine. This is not possible, as the Pevensie children were evacuated to the countryside at the start of the previous movie, which (as it featured an air raid on London) could not have happened before September 1940, and it is quite clearly summer during the scenes at Professor Diggory's house. A few minutes later Peter mentions they've been back from Narnia for a year, meaning that it is at least May 1942 now.
When the Pevensies are outside their treasure chamber in Cair Paravel, Peter tears off the entire bottom of his shirt, which would not have worked with a button down shirt because it splits in the middle, to make a makeshift torch. After Edmund takes out his flashlight, the camera once again shows Peter. The missing piece of his shirt is only on the right side. The rip changes again when they are entering the vault
At the beginning when the children are transported to Narnia on the beach, they are at first inside a cave. They run out of the cave onto the beach and start taking off some of their clothing to play on the beach. When the shot shows them from the other angle, the cave is now filled with water because the tide has come in to flood it - in just 5 seconds.
After the Pevensies save Trumpkin, he stands up and throws his gag on the ground. When he points to the river and states that the Telmarines were doing fine drowning him without their help, his gag is back around his neck.
Near the beginning of the movie when the children are saving Trumpkin they come up to the lake on the beach. The soldier's boat is in the middle of the lake. However, when Peter jumps in to save Trumpkin he jumps right above when he is thrown in and not from the shore.
When the Telmarines are about to throw Trumpkin in the river, Susan shoots an arrow that lands near the rear of the boat below the steersman. As the camera angle changes and the arrow is now more toward the front of the boat, under the rower.
Before Peter, Edmund, Susan, and Caspian head out for the final battle, Susan takes out an arrow from her quiver which she holds in her left hand. As she runs, the arrow has vanished and Susan is just starting to take out an arrow.
In the final battle, the Telmarines have invented a perpetual motion machine! That catapults have the lighter arm (after releasing the boulder) coming down to launch another boulder. Physically, the heavier arm would stay on the ground, preventing it from working.
The children play on the beach upon first returning to Narnia, then are in Cair Paravel in the next scene. The wetness and position of their clothing is different, but that is because much time has passed.
When the bear charges Lucy on the beach, Trumpkin seems to pull a bow out of nowhere. However, when Susan nocks an arrow to shoot the bear, Peter, Edmund, and Trumpkin are clearly seen running to the boat to grab weapons. It's entirely feasible that Trumpkin found the bow and arrow in the boat.
It's impossible for the soldier to catch up to Lucy. She wears no armor and weighs significantly less than the soldier in full metal armor. However, one would not be a good tactician, as the Telmarines thought themselves, by having just two riders going after them. These new two soldiers may have been patrolling, waiting for someone to try and make a break for it.
When escaping the castle, Caspian and Peter and the Professor were on horseback. When they are approaching their niche hideout, they were walking with no horses in sight in the background. What happened to the horses? (The girls used the big Friesian horse to escape the hideout to look for Aslan.)
(at around 40 mins) In the scene where catapults are being built, there is a two-person saw in use. In actual use, the worker "in the pit" below the sawn log should be covered in sawdust and wood shavings. However, the actor in the shot is clean.
The goof items below may give away important plot points.
At the end, when the Pevensies walk through the magic tree that will take them back to their own world, the three oldest siblings supposedly walk through, while Lucy waits right next to the trunk to take a final look at Aslan. Then it cuts to a front-shot of them walking through, but now Lucy is far away from the trunk, almost back where Aslan is standing. And the other three should have passed through already.
Towards the end of the sword fight between Peter and Miraz, when Miraz thrusts his shield towards Peter it is held by his left hand. He then shifts it so it held in his right hand, but a split second later the right hand is suddenly grasping the edge of the shield. This switch is continued when Peter twists the shield so that Miraz's left arm is pinned against his back.
When Sopespian calls for the troops to fall back to Beruna, they run toward the trees. However, when they reach the river, there is an enormous mountain directly behind them which they would have had to climb over.
At the end of the film when the children have gone back to the station, the boy Susan met earlier addresses her as "Phyllis" (she said it was her name), but he heard Lucy call her "Susan" clearly and the look on his face indicates he knows Susan has lied to him. However, there's no clear indication the boy clearly understood that Susan was clearly lying or if she was just caught up in what someone else said. The fight that was going on may have been enough for him to conclude she saw it and was going to watch.