Pilfering from Cheers :

[Patrick]

Sketches or situations that are in my opinion Cheers inspired:

In The Unkindest Cut Of All : Daphne's rejection of fitting owners for the puppy resembles a lot to Diane's rejection of buyers for the house that she has made Sam buy and then refuses to inhabit because she meets the previous owners even the passage where Frasier comes with drinks and appetisers is similar to that of Sam doing the same.

In A Lilith Thanksgiving : Frasier and Lilith's coming back, again and again until they get Fred's admission to a school is very similar to Sam and Diane's doing the same to a couple's counsellor( Dr. Simon Finch-Royce) who initially thought that they shouldn't be together until he cracks and declares (I found the exact quote) :

"Hear this, world. The rest of you can stop getting married. It's been done to perfection. Envy them, sofa, envy them, chair, for you shall never be as cozy as they for their union shall be an epoch-shattering success and I STAKE MY LIFE ON IT!!! Wait a moment, let me get this on record. I, Dr. Simon Finch-Royce, being of sound mind and body declare that Sam and Diane shall be happy together throughout all eternity and if I am wrong I promise I will take my own life in the most disgusting manner possible. Here, take the tape, NO, take the whole machine. It's my wedding gift to you. The most perfect couple since the DAWN of TIME!!!! " (I love this passage!)

When I get the time I'll post more of these but if you think of some then don't hesitate to contribute.

[Frank B]

There was one episode in Cheers where Frasier threatened to jump from a ledge, and one episode in Frasier where Bebe did the same thing.

[Patrick]

True.

I have another one:

They already did the Frasier playing Chess again and again and losing every time and obsessing about it, except that in Cheers it was Woody (of all people) and not Martin that kept beating him. As far as we know, Frasier never beat Woody.

[DiceMan]

Not really a sketch, but there's a bit in Cheers where Rebecca whispers seductive stuffs to Sam at a restaurant and then gets up asking him to follow her to which he says (still sitting down) "Could you just give me a minute"

I just remembered Niles doing something similar in the one where he's giving Daphne piano lessons

[Patrick]

Not really a sketch, but there's a bit in Cheers where Rebecca whispers seductive stuffs to Sam at a restaurant and then gets up asking him to follow her to which he says (still sitting down) "Could you just give me a minute"

Yes, I remember that episode, he's managed to get her her job back as manager of the bar but in exchange of drastic concessions on her part, like a reduced salary and the fact that she has to obey orders from Carla. She thinks she got her job back without conditions and that's why at first she is grateful and agrees to have dinner (and maybe more) with Sam but when she learns the truth thanks to an allusion from the big boss, she decides as a way to get revenge to arouse Sam and let him at the restaurant, incapable of getting up without publicly embarrassing himself.

I just remembered Niles doing something similar in the one where he's giving Daphne piano lessons

Yes Niles just had a passionate kiss with Daphne on the piano bench and we imagine that he must have something that impedes somehow his movements for a while.

[Frank B]

There are also the two wifes, Vera and Maris, whom we only hear about, but never see.

[Patrick]

Yes, and Carla's family is somewhat similar to Daphne's family. Woody's kooky stories and Daphne's.

[Frasiertime]

I came across this one in some trivia, thought it kind of interesting:
◾In "The Harassed" (2003) Frasier and Julia Wilcox are having a heated argument, during which Frasier asks Julia "Are you as turned on as I am?" In the Cheers episode "Showdown - Part 2" Sam and Diane are having a similarly big argument. During this argument Sam asks the exact same question, to which Diane replies "More!" and they kiss. The episode aired in 1983, and as an in-joke Bulldog later criticizes Frasier's rather dated choice of line by saying that no one has used the term "turned on" for 20 years.