The scene where the couple who would eventually become Teddy's parents are driving down the road with Moe is supposed to be 25 years prior to present day (2012), yet following behind them is clearly a red Jeep Liberty, which didn't go into production until 2001.
Ted refuses to give the trio the $830,000 needed to save the orphanage because he blames the orphanage for the things that went wrong in his life; from being sent to military school to his murder plot. One, the orphanage could never predict such things would happen at the time he was adopted, and two, Teddy seemed to not have any problems coming to that exact same orphanage later with Ling to adopt Murph, Peezer, and Weezer.
After the car explodes in the lake, Teddy and the Three Stooges are talking on a bridge. Teddy's face is initially dirty with a black substance (mud? soot?). As they talk and the scene continues to cut back to Teddy, his face successively becomes cleaner until there is very little dirt on his face. Then, at the final cut of the scene, his face is completely dirty again.
Teddy says he would not be able to open the door to the car they're all in because there's too much pressure from the water on the outside. While that may be true when there'e no water in the car, once the water pressure evens out, the door has a better chance of being opened. Also, no one suggested trying to roll the windows down, which even though were power windows, might still roll down in the early stages of the car being in the water.
At first glance one might think it's a mistake that many of the nuns didn't age at all during the course of the movie. It's more likely that it's all part of the joke since the original Stooge shorts were made on the cheap and the directors would have never bothered with details like that.
When Larry and Curley reunite with Moe on the Jersey Shore set, the producer who got Moe on the show is surprised to see there are three of them, conveniently forgetting he was watching all three together on stage before Larry and Curley walked off and he gave Moe the part.
In the baby-changing scene, in which the trio is using diaper-less babies to hit each other with their urine, you can clearly see a diaper on the baby Moe is holding when he goes "Dirty Harry" on Curly.