Deliberate mistake: Towards the end of the movie fish are seen to swim down in order to submerge the net in which they have been caught. Although the attempt seems logical, it actually needs power against the net to push it down. None of the fish seem to touch the net at the bottom however and if it is the whole mass of fish achieving it, then the bottom fish should have been squashed against the net which (again) isn't the case. 'What brings the net down?' is the question here. (Done deliberately by Disney/Pixar to be visually aesthetically pleasing.)
Continuity mistake: In the sequence with the boat, its position changes. When Nemo is looking at it with his friends, it is a long way away. When Nemo gets angry and swims to it, it's suddenly very near the reef. Then, the diver catches him when he's only been swimming away from the boat a few seconds, but the diver has to swim a long way back to the boat. Finally, when Marlin follows the boat, it's very near again, because he gets there before it starts, even though he was temporarily blinded by the camera. The boat does all this without raising the anchor until it leaves.
Continuity mistake: Towards the middle of the movie, we see a shot with Nemo staring out of the tank, depressed. The camera slowly pans toward the side of the tank and Nemo is reflected in the glass twice. However, towards the end of the movie when Nemo is playing dead, all the fish swim to the other side of the tank and Bloat says something like, "What's happening? Why is he playing dead?" The camera pans to the side of the tank again, but the fish are only reflected once this time. They should have been reflected twice.
Deliberate mistake: When Dory is inside the whale, she doesn't have the scars from the jellyfish sting anymore. [This is actually a long-standing Disney tradition. The company seems to believe that it is less traumatic to see violence onscreen if the evidence is gone immediately (blood, scars, etc. always disappear). Still an error, but it wasn't the fault of the CGI animators.]
Factual error: Throughout the movie, to make it more authentic, small impurities (specks) are seen floating in the water. Throughout most of it, with great attention to detail, the impurities are always seen to be moving with the flow of the water - back and forth. This always works except in the first scene with Marlin and Coral. The sun is shining through the water, and you can see the anemone waving in the water flow, but the impurities are all flowing in one direction, left to right, never changing according to the water movement.
Factual error: After Nigel returns Marlin and Dory to the ocean it is raining and the sky is nearly solid with gray clouds. As Marlin and Dory speak bright sunlight is streaming down through the water all around them, illuminating the ocean floor. Even if the water is extremely shallow where they are swimming the overcast sky would not provide such bright, consistent light.
Factual error: At the end of the movie it shows all of the aquarium fish in the ocean inside of their bags. It also shows the bags almost completely out of the water with only a small portion under water while the bags are almost completely filled with water. With this amount of water the bags should be submerged underwater with only the little amount of air showing on the surface, because water is not lighter than water; water weighs the same.
Factual error: The film is explicitly set on the east coast of Australia. However Nigel is a Brown Pelican (Pelecanus occidentalis), a species only found in the Americas. The only pelican species found in Australia is the Australian Pelican (Pelecanus conspicillatus) which is mainly white, with black wing tips. If a Brown pelican was somehow transported to Australia and subsequently escaped into the wild, it would have been caught and destroyed under Australia's strict quarantine laws.
Plot hole: When Nemo swims off to the boat, a diver pops up and catches him, and all the fish react. But this diver should have been visible to Marlin and co. long before it reaches Nemo, as they are far enough from the boat and looking out into open sea, yet none of them reacts until the diver is right behind him. Also, bubbles should have risen into shot before the diver does.