What form of discourse is Dr. Martin Luther King, Jr.'s "I Have a Dream" speech?

King applies several forms of discourse in the speech, which is part of what makes it so effective. In addition to argumentation, for instance, he also uses narrative discourse.
Narrative discourse accounts for events in history and explains how those events have an impact on our present circumstances. King invoked the legacies of the Declaration of Independence and the Emancipation Proclamation to tell the story of a nation committed to freedom, but repeatedly failing to deliver that freedom to its black citizens.
His dream is, indeed, constructed from a historical narrative. When he talks of the day when the "sons of former slaves and the sons of former slave owners will be able to sit down together at the table of brotherhood," he is reversing past customs that would have forbade such socializing to advocate a more equal future.
Narrative discourse does not only use history, but applies origin stories, customs, and personal stories to show that actions "are contingent on one another"—that is, events which occur now are due to those which have occurred in the past.
https://www.archives.gov/files/press/exhibits/dream-speech.pdf

Martin Luther King, Jr.'s "I Have a Dream" speech is the form of discourse known as Argumentation.
In an argument-driven piece, speakers present evidence in order to convince their audience of the logic of their points of persuasion on a certain topic. In other words, the speaker urges his listeners to agree with his points so their minds change and positive action occurs. 
Certainly, change and action are what Dr. Martin Luther King sought when he delivered his speech in Washington, D. C. Certainly, too, his style of delivery was persuasive in his mention of injustices and in his tone of voice and use of rhetorical devices.
One of Dr. King's arguments is his reference to the United States Constitution in which it is stated that all men are guaranteed the "unalienable Rights" of "Life, Liberty, and the pursuit of Happiness." Using the metaphor of "a promissory note" for this promise of the Constitution, King argues that the "Negro people have received a bad check," meaning they have not been offered the same opportunities as other American citizens. He cites segregation as the main prohibition of opportunities for African Americans. He argues the promises and ideals of the Constitution can only be mandated through an equivalent promise and commitment to the cause of Civil Rights.