When in the hotel room, while Frasier is reading aloud the first sentence Niles wrote, his mouth does not move when the camera switches to a side angle, even though he continues to read.
When Martin begins to tell Frasier and Niles his anecdote about his first partner, he is named Mitch Gosset. A few moments later, Martin and Daphne refer to him as 'Goss', which people are mishearing as 'Gus' and think it is a continuity error. In fact, in the initial anecdote Martin also refers to him as 'Goss' (as a nickname, a short form of his surname) so it is not actually an error.
During the scene where Frasier (Kelsey Grammer) and Niles (David Hyde Pierce) begin to write in Frasier's apartment, it is clearly visible when the shot cuts to Niles that the screen to his laptop isn't on while he is shown to be typing.
The view from the hotel window is clearly a painted backdrop.
When Martin is lecturing Frasier and Niles about brotherly love he tells them that even though he never had a brother he did have a falling out with a partner. However in Season 5s Beware of Greeks (1998) he not only has a brother but he has two, and they are estranged from each other.