In Chapter 7 of The Great Gatsby when Nick says that "in this heat, every extra gesture was an affront to the common store of everyday life", I know that he is referring to how the butler's actions but what, if any, connection does this quote have to the theme of the novel (moral decay, wealth, and the American Dream)?
This is a good and challenging question. If we presume that this line is reflective of the larger themes of the novel relating to wealth, opulence and moral decay, we might argue that the quote suggests a division between the "common/necessary" and the "uncommon/excessive".
Many of the novel's characters and settings express the idea that wealth results in moral abuse and financial waste. We can see the "waste" in examples like Myrtle's purchase of a dog for an apartment she where she does not live and Gatsby's lavish parties where he does not speak with his guests.
We might say that this line, ostensibly concerning the heat, refers also to the repeated distinctions between the rich and the poor as the rich often pursue impractical and unnecessary luxuries as a way of demonstrating status.
Going beyond the practical and fulfilling fanciful dreams, the wealthy class in the novel go out of their way to prove they are "uncommon" in a class-oriented sense. There is a notion that money and resources are inexhaustible for people like Tom and Gatsby.
Perhaps we see in this line about the "common store of everyday life" an implication that this sense of inexhaustible resources is a false one. There is a necessary limit to anyone's resources and a limit to dream fulfillment as well.