Cory says that he and Shawn are 12 years old, yet in the very next episode ("It's a Wonderful Night"), Cory's father says that Cory is 11, and in the episode after that ("Kid Gloves"), Cory has a birthday. Since he's in 6th grade he's probably turning 12.
When Alan is talking to Cory in the scene with Shawn under the bed, he is using his left hand to emphasize his point. In the next shot however, he is using his right hand.
In the scene right before the opening credits, Cory's door changes from being closed to open to closed again between shots.
When Alan is coming up to talk to Corey in the bedroom, Shawn slides under the bed headfirst from the foot of the bed.
After Alan leaves, Shawn comes back out headfirst (also from the foot of the bed), even though he was never able to turn around.
After Alan leaves, Shawn comes back out headfirst (also from the foot of the bed), even though he was never able to turn around.
After the power is shut off Eric complains that the phone died. A land line phone as long as the battery is charged will still work if the power goes out since it does not need electricity to work.
At the end, when the power goes out, Amy says "I didn't save!" However, she is on a laptop, so even when the power went out, the laptop would still have power from the internal battery.
When Amy and Morgan enter Cory's bathroom, Shawn somehow gets out (presumably through a window). Although in "Hair Today, Goon Tomorrow," the bathroom is shown, and there is no window, Shawn could have escaped through Morgan's bedroom window since the bathroom connects the two rooms. Amy mentions this when she says that Morgan's door to the bathroom is locked.
Alan says Cory and Shawn have been best friends since Shawn taught Cory how to pick the lock on his playpen. But the flashback in "It's Not You, It's Me" shows Shawn and Cory meeting for the first time at the zoo, aged around 6 or 7. Alan is speaking figuratively to say Cory and Shawn have been friends for a long time and not referring to literal events.
When Mr. Feeny is reading "The Tell-tale Heart" to the class he skips a good portion of it ('They were making a mockery of my horror!-This I thought, and this I think. But anything was better than this agony! Anything was more tolerable than this derision!') before returning to what the actual text is.