Gatsby offends Nick when he offers monetary rewards for the kindness Nick has shown to Gatsby. Gatsby does not understand that Nick doesn't want 'things' in exchange for friendship. The meeting between Gatsby and Daisy has an adolescent teenage mentality to it. Gatsby believes that he can prove his love for her by a guided tour through his mansion, it is a classic case of 'boy tries to impress girl'. All the wealth Gatsby accumulated was in pursuit of one thing, his romantic dream that he and Daisy would be together. It never happened...
Gatsby asked Nick if he wanted to come work for him, which offended Nick because Gatsby was offering out of concern for Nick's well being. In other words, because Nick had a small house and didn't have the lavish lifestyle that he enjoyed, Gatsby felt Nick was "in need" of his help.
When Gatsby and Daisy met, it was awkward at first. Gatsby ordered his servants to set up tea with the finest wares, and even brought in flowers by the garden-full to decorate Nick's cottage. Everything Gatsby did said that Nick's place wasn't "good enough", but this was because Gatsby has always thought and continues to think the way to Daisy's heart is through superficial monetary gain and material possession. Unfortunately, Daisy only serves to perpetuate this opinion, as she literalyl cries when she sees Gatsby's well-folded shirts.
In Chapter Five, Gatsby offers Nick an opportunity to earn some extra money and it is this offer which causes some offense. The details of this offer are not particularly clear: Gatsby says that this opportunity is a "confidential sort of thing" that would not take up much of Nick's time. But Nick immediately cuts Gatsby off by claiming that his "hands are full" with his job selling bonds.
To understand why this offer causes offense, we need to look at Nick and Gatsby's previous conversation. Gatsby asked Nick to arrange for him to spend some time alone with Daisy and Nick has agreed that this will take place in two days' time. It appears to Nick, then, that Gatsby's offer is a payment for arranging this meeting. Nick acknowledges this in the text:
"But, because the offer was obviously and tactlessly for a service to be rendered, I had no choice except to cut him off there."
For Nick, then, it is inconceivable that he would accept money for arranging a meeting between his cousin, Daisy, and Gatsby because this implies that she is a commodity to be bought and sold.