The Great Gatsby is definitely not a drama because it is not written in the form of a play. As far as historical fiction, the setting is crucial to the characters and events in the story. The setting also needs to mark a particular period of time in which an event actually took place, such at The Crucible by Arthur Miller, which is both a drama and historical fiction. Although the setting is important to the characters and the events in the story as far as the "traditional rich" vs the "new rich," that is the extent of it. It depicts more of the people of the time than a historical marking in time. Therefore, The Great Gatsby is not a drama nor historical fiction. It is simply a piece of fiction.
No, I would not classify "The Great Gatsby" as either a historical novel or a drama. If I had to choose one of those, I guess I'd choose historical novel, as dramas are usually plays and Fitzgerald's work does capture the feeling of the Jazz Age well. However, if I were allowed to choose more freely, I'd say it is a blend of Modernism and realism. The portraits of society are nicely realistic, as is the social critique, but the symbolism is nicely Modernistic, and sum up the inner lives of these characters well.