Burr will remain a Democratic-Republican in good standing and will have a reasonable chance to succeed Jefferson as president in 1808. I don't think the duel with Hamilton takes place, but I think Hamilton and the Federalists will be a spent force anyway. There might not be a Twelfth Amendment--the Democratic-Republican party might settle on counting on a couple of electors to vote for someone other than the party's vice-presidential candidate; after all, that will have "worked" in 1800.