James gets out of bed in his nightshirt, fights his aunt and falls down the stairs head first. His night shirt slips up and reveals that his stunt double is wearing white pants down to his knees. The next shot of James at the foot of the stairs shows that he's not wearing pants.
When the peach goes rolling down the hill, it rolls along the top of a fence, getting the fence stuck in its sides in a counter-clockwise descending spiral. Once the peach hits the water, though, every scene after shows the fence in a clockwise descending spiral, when viewed from above.
When Spiker and Sponge are in the car attempting to drive away from the peach Spiker is the driver on the left side. She should be driving on the right side of the vehicle as they live in the English countryside.
There is no way James's aunts could have driven across the ocean from England to New York. Cars can't float on water and one would not survive in a submerged car. Even so, it's impossible to drive at a comparable distance in the time it took for James and his insect friends to travel to New York.
When the Centipede, the Spider and James are trying to get the compass from the pirate ship, the Centipede pulls the string twice to signal to the other bugs to pull them up. But the Centipede was not there when James told the other bugs to pull them up when he yanked on the sting twice.
At the start of the movie, the narrator states that a rhino appears out of nowhere and eats James' parents. The aunts bring up the rhino several times during the movie. It's later revealed that the rhino is just a bunch of smoke and noise, which makes no sense because that same rhino ate James' parents. Perhaps the biggest error of all is the fact the whole thing with the rhino "appearing out of nowhere" is never explained. In the book, James' parents are eaten by a rhino that escaped from the London Zoo while they were shopping at the market. The book explains it, whilst the movie does not.
After James looks at the map and figures out that they are going to New York City, the Centipede spins his Cigar in reaction. Two filament lines attached to the Cigar that were used to suspend it are clearly visible.
The goof item below may give away important plot points.
When James wakes up from the fall onto the Empire State Building, he is inside the peach, with a red light flashing on him. But he has to open the hatch in the peach to get atop it, so, the light shouldn't have illuminated him at all, as it is from a light above the peach.