It's implied that Frasier is the subject of Daphne's psychic vision of a man "wearing wingtips and a trench coat... getting off an elevator... walking down a long hallway... [and] opening the door." But Frasier's apartment is directly across from the nineteenth floor's elevator, so he wouldn't have walked down the hallway between getting off the elevator and opening the door.
As soon as Frasier gets up from the piano after having been stared at by Eddie the Dog, when the camera cuts to a wider angle, Eddie has disappeared.
When Niles shows Frasier the next day's newspaper in the Cafe, it is the same paper he showed Frasier the day before because the picture on the back of the newspaper is the same picture as the previous days.
Martin says he has been trying to solve the murder of a prostitute that happened 20 years earlier. Later he says the murder occurred when Frasier was 12 years old. Frasier would have been much older than that at the time.
When Frasier launches into his in-studio diatribe against Derek Mann, he removes his headphones. A broadcaster always keeps his or her headphones on when they are on the air.