Oh, I think that Burr as the 4th President is a little too obvious for the devious Mr. Burr. Jefferson hated him, and Burr had just lost the New York Gubernatorial race.
When he initially moved West (post-Hamilton and post-VP), it was thought that he was trying to reestablish his political career in climes more agreeable to things like dueling, where his "murder" of Alex Hamilton would be viewed in more agreeable terms.
The man was very intelligent and very political and a very, very good political organizer (he started Tammany Hall, 'nuf said). If we avoid Hamilton's death, I think Mr. Burr moves West for real, and runs for Senator in Kentucky or Ohio, or maybe somewhere more southern, Tennessee or Alabama perhaps? He lived for an awfully long time, and I think that he would have ended up as quite a political player in the West.
He would have been a war-hawk in 1812, and I think that he would have been a Jackson supporter in 1824, and ended up as a major player inside Jackson's nascent Democratic Party.
Really, I think that he would have ended up as an important player in the political process, seen by some historians as something of an opportunistic Democrat, and by others as the living bridge between the Revolutionary generation and their immediate post-Revolutionary successors. I think that he would have been at least as important as Henry Clay in national politics and the history the United States in the first half of the 19th century.
I don't think that he ever would have been able to run for national office again, his attempt to gain the presidency in 1800 was too much of a stain to ever erase. That doesn't preempt him from playing an important role in the formation of the the Democratic Party (after all the Party was just a collection of state machines, and Mr. Burr certainly would have been a machine boss by the time Jackson's Presidential bid gave them a reason to unite).
His daughter married into a prominent South Carolina political family (his daughter's husband became Governor of South Carolina), and I think that Burr certainly would have left a political legacy for his family to inherit. He died in 1834, and if his first grandson had lived he would have been 32, and probably already the heir apparent to his grandfather's political machine and legacy.