Stupidity: When Donatello frees the professor, he gets his head in his visual range, and same goes for his thick inhuman hands, but he notices only when the plot says so, a few moments later. (00:52:50)
Suggested correction: The professor is in noticeable distress, likely from having been tied up for so long and hearing the fight between the foot soldiers and the turtles occurring just outside, so he isn't paying much attention to who is rescuing him at first; he just wants his hands untied and to get the tape off his mouth.
Stupidity: The Foot Clan knows that April O'Neil is an ally of the Teenage Mutant Ninja Turtles. Since the Foot Clan hasn't been able to find the Teenage Mutant Ninja Turtles for where they live a easy solution would be for them to go to April and ask where are the Teenage Mutant Ninja Turtles. An explanation is never given in the movie for why the Foot Clan never use that easy solution.
Suggested correction: First of all, the Foot Clan at the beginning of the movie is disorganized. Secondly, upon Shredder's return, he plants a Foot soldier named Freddy into April's news crew in order to keep an eye on her and find out where the Turtles are. Then they change tactics when they discover there is still some of the mutagenic ooze around, which they then decide to use to create their own mutants to lure the Turtles out of hiding.
Stupidity: When the Turtles are leaving the junkyard they rescued Raphael from, the incredibly agile ninjas of the enemy clan can't simply jump over a van, or just slide over the hood. (00:54:00)
Suggested correction: Remember at the beginning of the film, one of the clan members is angry that so many of their members were arrested. They have been looking hard for new recruits. The new recruits won't be as good right away.
I am sure even my mom could slide across the hood of a parked car if need be.
Stupidity: Leonardo could use his swords to cut the net when Leonardo, Donatello, and Michelangelo are trapped in a net.
Suggested correction: While trapped in the net, it would have been quite difficult for him to draw his swords from the sheath on his back.
While that is true, him not attempting to reach his swords makes him look like a fool that forgot he has swords.
Plot hole: It's never really addressed or explained how Shredder survived his death in the previous movie from being crushed in a garbage truck.
Suggested correction: It didn't really need to be addressed, and thus is not a mistake. His hand emerging from the junkyard is enough to show that he survived his apparent demise at the end of the last film.