During the briefing on the carrier before the Doolittle raid, a background character is wearing two parallel vertical metal bars on his overseas cap, which would normally indicate an army captain rank, a commissioned officer, except the bars are gold when they should be silver. On top of that, he's wearing sergeant's stripes, which is non-commissioned officer (enlisted) rank. He can't be both.
As Danny and Rafe try to take off from the airfield, three enemy aircraft are closing in on them, guns blazing. In the first shot, the aircraft are D3A1 "Val" dive-bombers (distinguished by their fixed landing gear in bulky fairings); in the next shot, however, the aircraft are replaced by A6M2-21 "Zero" fighters, with retractable landing gear.
When the planes take off to bomb Tokyo, the captain of the carrier says "forward" in order to help them taking off. When the planes are still taking off, the carrier has its anchor chains as if it was anchored. In reality, the filmmakers were working on the USS Lexington (CV-16), a carrier-turned-museum that was indeed docked.
When the major brings Evelyn into the command post area, he tells her Doolittle's raid is scheduled to take off in a "couple of hours". If the task force was spotted by a Japanese ship and immediately launched which advanced the raids execution by 12 hours (as reported to President Roosevelt), how could the Major have brought Evelyn to the command post a couple of hours before they were scheduled to be launched. The raid would have been over.
Crew or equipment visible
Errors in geography
Throughout the attack scene, the placement of the battleships in Battleship Row changes. When the Oklahoma is shown capsizing, in some scenes she is correctly moored next to the Maryland, in other scenes she is next to the destroyed Arizona. In some parts the capsized ship is even surrounded by some sort of fog with no ships around her. When Admiral Kimmel is on the small boat touring the harbor after the attack, the Oklahoma is next to the Arizona and other battleships that seem to have been placed in a random clutter next to each other, instead of the line that they were in that morning. Even in the scene showing Pearl Harbor at an aerial view right before the attack, the Geography of Ford Island and the placement of the battleships is wrong. Battleship Row isn't even visible.
After the Doolittle raid, the Raiders head west to China. In one shot, however, the Raiders are shown heading away from the sun as it sets behind them, which means they are heading east and therefore back to Japan. This would be very unlikely considering the dire fuel situation on board the aircraft.
In the scene where President Roosevelt is expressing his dismay with the Americans not doing more to aid the Aliies in Europe he mentions that the US needs to send more tanks to Britain and Russia, to provide aid. At this point in time, early 1941, Russia was still an ally of Nazi Germany. They didn't start fighting on the side of the Allies until after the invasion of the Soviet Union in June 1941 and were not considered part of the Allies until January 1942.
When the Japanese patrol boats are contacted, it is said that they are 400 yards away. At a distance of 400 yards ships at sea are in danger of colliding, not just being spotted. The actual sighting was made at 10,000 yards. Further, such a sighting was no great surprise, since an air patrol had spotted Japanese ships earlier that morning.
It would have been virtually impossible for Rafe to have served with the RAF. The movie makes it almost seem like some sort of secondment but, in actual fact, the US Army and government had a lot of rules and laws in place to prevent their serving airmen from flying with foreign air forces. The AVG Flying Tigers in China, for example, and the American Eagle squadrons in RAF service were manned by men who effectively broke military law and had to go by very circuitous routes to get there. Rafe would have had to resign his commission and work out how to get through a war zone as a civilian to end up flying Spitfires in Europe. Somehow he repeats the feat in reverse, turning up back at Pearl with his rank restored - very unlikely, especially when the USA was neutral and very hostile to the idea of its servicemen flying with other combatants. He more than likely would have been court martialled.
Incorrectly regarded as goofs
The Japanese are shown flipping a calendar from the 6th to the 7th of December on the morning of the attack. This is done for American audiences who are familiar with the date of the attack being 7 December 1941. Clocks aboard the Japanese ships were kept on Tokyo time, so for them the attack actually took place the morning of 8 December. The Japanese version of the film shows the calendar flipping from the 7th to the 8th.
American pilots serving with the RAF were not released from service until after the U.S. entered the war as an ally. It would have been extremely unlikely for Rafe to be back in U.S. service before Pearl Harbor unless he deserted from the RAF (in which case the U.S. Army Air Corps would not have accepted him).
Rafe has dyslexia and when he fails his eye exam, he explains to Evelyn that his vision is excellent and that he is a skilled marksman back home - he just has difficulty telling letters from one another. However, he tried to fake his way through the eye exam by looking at a crib sheet with the eye chart reproduced on it. Since he cannot tell letters apart, if he cannot read then eye chart, then he also should not have been able to read the crib sheet.
When Rafe and the other pilots are attempting to reach their planes, he tells them that P-40s can't outrun Zeroes, but can out-turn them. Not only is this information untrue, in fact the truth is quite the opposite, but how would he know this information? The Zero was only introduced in 1940, and very few, if any, Americans had ever seen it in action before Pearl Harbor.