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.] (01:09:30)
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.)
Deliberate mistake: Sydney's real harbour skyline has some neon signs on top of buildings. In the movie all them are omitted, presumably for advertising and aesthetic purposes.