At about 37:58 when Shredder tells Tatsu to remove the bar because "it's time they learned who their master is", the shot goes back to show Tatsu from the front, where his lips are clearly moving, but there is no sound.
During the junkyard scene, just before Donatello is thrown, you can clearly see that Leonardo's katanas are in their sheaths. After he is thrown the shot cuts back to Leonardo fighting and his katanas are no longer there.
At about the 52:47 mark after Donatello is thrown through the roof and rips the tape off of the TGRI guys mouth he exclaims "I gotta get the hang of that" to which the TGRI guy says "That's okay." immediately after "That's okay" you can hear someone on the set say "Teenage Mutant Ninja Turtles 107" in the background. Use headphones if you must.
In the first fight scene, when the turtles come down the stairs behind Keno, they come to a stop. Before they jump into the air for the title screen, you can see them jump onto what is assumed to be a trampoline.
When Keno is waiting outside for the Foot audition, a certain part of Manhattan is visible in the background, however, when he is walking into the building, the background shows that he is at least a mile away.
Tutso asks Shredder, "We rebuild the foot?" Shredder specifically says, "No." (because the only one thing that is next is revenge.) Later, Keno tells the Turtles that he's heard of the foot, because the word is out on the street that they are looking for anybody with martial arts talent, especially teenagers. That means they are recruiting people to join the foot, thus meaning that they are rebuilding the foot clan, even after the shredder said that there was no need to do so.
When Keno is running from The Foot's Soldiers in the junkyard aka. The Foot's HQ, you see him fighting against two of The Foots soldiers, and another guy running in the back, ready to do jump over the junk in front of him, and do a saltomortale. Before the saltomortale, he jumps up into the air, like he is about to jump onto a trampoline.
Before the final fight with Tokka and Rahzar, when Donatello and Professor Perry deduce that they need CO2 in order to destabilize the mutation, they find a fire extinguisher. When they grab it, you can see from the pressure gauge that it is in fact not a CO2 extinguisher. This would make sense because CO2 extinguishers are primarily used for closed in areas or electrical equipment.
The goof item below may give away important plot points.
When Shredder raises his hand out of the dock that has collapsed on him at the end, it eventually falls limp indicating he is dead. However, the spikes on his wrist shake, indicating that they are in fact rubber.