Deliberate mistake: Shawn fishes Mr. Feeny's watch from the septic tank and mentions the septic tank is deeper than it looked. Given this, he should be covered in human waste, instead of just looking a little dirty.
Better than Average Cory - S6-E5
Deliberate mistake: When Rachel is choking and writes down "I'm choking to death" on the paper on the counter so Jack will help her, you can see she is just scribbling the pen across the paper and not actually writing anything. (00:06:36)
Janitor Dad - S4-E6
Deliberate mistake: Chet comes home from his first day as school janitor brandishing a paycheck, which impresses his wife and in turn changes Shawn's mind about Chet working at the school. Being that it was Chet's first day on the job, there's no way he would already have received a paycheck. Even if it was payday, his employment would have begun on the first day of the next pay period.
Suggested correction: It's possible it was pay for training or orientation days that were before his official start date.
The Last Temptation of Cory - S3-E9
Deliberate mistake: When Cory calls Topanga from Missy's phone, he starts having a conversation with her as soon as he finishes dialing her number, with no time for the lines to connect.
Deliberate mistake: When Alan is trying to rush Amy out of the door, she yells from upstairs that she's still shaving her legs. A mere 20 seconds later, she comes downstairs wearing a dress and nylons. Even if she was for some reason shaving her legs while wearing a cocktail dress, that still wouldn't be enough time for her to clean off her legs, put nylons on and then walk downstairs.
Deliberate mistake: Morgan turns on the stereo and starts dancing on the sofa with her doll. She then tosses her doll, which tips over a plant and breaks the glass on a set of French doors. The stereo magically stops the moment the glass breaks.