In some scenes, the dialogue for several characters is written as play dialogue. Why is this?

This novel, perhaps more than any others written by Fitzgerald, closely parallels his own life as he struggled to make ends meet, despite being one of the highest paid authors in the country. The financial collapse came at the end of a decadent period in history, and that is portrayed in the novel as well. This stylistic approach described in the question is done to show that the main characters, Anthony and Gloria, feel like their lives are playing out separate from them. They're waiting for an inheritance that will make their lives easier, but it is slow in coming, and yet they still want to continue the extravagant lifestyle they had when they first got together. The reality of poverty begins to wear away at them, making them cynical and depressed. Writing some scenes of dialogue in the style of a stage play underscores the surreal feeling that Anthony and Gloria both have about their situation: they feel like they are "performing" in front of their friends and social acquaintances as a couple that has plenty of money and is happy and carefree, but in reality they're destitute and unhappy.