The significance of the butler's nose is that it is because of his nose that he has his current job. He used to be a silver polisher for a family in New York. But it started to bother his nose until he had to give it up.
To me, this indicates that the Buchanans are not as high class as they seem to be. Their butler used to be a lowly silver polisher. That is a much lower job than being a butler and so it seems that the Buchanans are willing to put that kind of a low class person in a high class job. That means they must be kind of low class themselves.
My personal thoughts:
They mentioned it to Nick just as a gossip. Evident from the text shows that Daisy was speaking to Nick in a semi-serious manner, yet talking about someone's nose in a serious manner is itself not serious. To me this only shows the gossiping of the females in the text; Fitzgerald doesn't portray women in a very favourable manner in this novel, and this is one of the ways he portrays women, by the gossiping of Daisy and Jordan.
The girls begin gossiping about the butler's nose in order to change the subject during Tom's racist rant about keeping the white race in power.
Both of the women try to change the subject. Daisy tries to mock Tom when she says "we ought to beat them down" as she winks at the sun. She is mocking him because she has already teased Tom about reading very "profound" books with "long words in them." You can tell Tom is not amused because he interupts her to continue his rant. Then Jordan tries to speak about living in California but is interrupted by Tom who continues his rant.
Then Daisy finally gets her opportunity when Tom is distracted by the rining phone. In the text it states "Daisy seized upon the momentary interruption." Then she leans towards Nick and begins the gossip about the butlers nose.
Notice that Nick mentions the butler's nose in a later chapter when Daisy comes to his house. Daisy doesn't have any idea what he is talking about, so Nick changes the subject. This shows that Daisy just makes random things up, like a child, but they mean nothing to her. Nevermind that it is gossip about another human being. Everything Daisy does is for her own entertainment or in order to avoid reality, and the consequences don't register.
Also, their is a nose motif here. In chapter two, the face of Dr. TJE on the billboard is described as having a "non-existent nose."
There is also a focus on Meyer Wolfsheim's nose, as well, and this is probably a play on the Jewish stereotype.
All of this makes me think that noses have something to do with money...Having a big one means you can sniff out a buck, and not having one means you are bankrupt.
This isn't a fully fleshed out idea yet, though, so don't quote me on it.