Episode I: the Baby Menace (3) - S5-E1
Revealing mistake: When Tommy is asking if everyone remembers which baby came from which incubator, you can see the baby he his holding in his left arm is just a doll. (00:09:10)
Continuity mistake: Dick's plot in this episode is that he learns about the concept of tipping. However, the season 3 episode "Eleven Angry Men and One Dick" showed that Dick was already familiar with the concept of tipping when he tipped a bailiff during a jury selection process.
Dick Puts the Id in Cupid - S5-E11
Other mistake: While talking to his therapist, Dick tells her that one of the mishaps that he caused in his relationship with Mary was showing up to a fundraiser naked. While true, this doesn't happen until Episode 13 (Rutherford Beauty) - two episodes later. This was due to the fact that most of Season 5 was aired out of production order.
Suggested correction: The show was produced in the right order, so this isn't a mistake. Being aired out of order isn't a mistake of the show. It should be noted that for each of the 6 season, episodes were constantly aired out of order.
Dick Puts the Id in Cupid - S5-E11
Factual error: At the police station, in lock up, there's a paper on the bulletin board with a phone number "1-800-VICTIM" An 800 number should have 7 letters/numbers instead of just 6.