Again, I'll quote an old soc.history.what-if post of mine:
***
In 1800, the Republicans did show some concern, but not much, about a
possible Jefferson-Burr tie in the Electoral College. The reason they
were not too worried is that they expected that one or two of their
electors would vote for Jefferson and some candidate other than Burr.
This expectation, alas, was based on rumor and conjecture and proved to be
unfounded. Of course, the Republicans could have made sure that Burr
would get fewer votes than Jefferson, but they seemed more worried that
Burr would get too few votes than too many; they wanted to preserve the
Virginia-New York alliance, and for that reason the electors from Virginia
(all of them Republicans, thanks to the legislature replacing district
voting for electors by at-large voting [1]) were strongly pressured to
vote unanimously for Burr as well as Jefferson (the Republicans remembered
that in 1796 too many Virginian and other Southern Republican electors had
"cut" Burr). The result of course was the Jefferson-Burr tie which sent
the election into the House.
Suppose the rumors had been true and one or two Republican electors did
vote for someone other than Burr? Consequences:
(1) No election in the House, obviously.
(2) Less presure to enact a Twelfth Amendment--though the closeness with
which an election in the House was avoided and the fear of Federalist
mischief in the Electoral College in 1804 (see below) may produce one
anyway.
(3) Burr, instead of being in disgrace for having flirted with the
Federalists to gain the presidency, remains an important if controversial
Republican. He is likely to be elected Vice-President again in 1804 and
to be seen as a possbile successor to Jefferson in 1808 by those who think
that there have been enough Virginian Presidents. (But he has the problem
in 1808 that the "anti-Virginians" are not united; there is the rivalry
between him and the Clintons in his own state of New York.)
One possible problem in 1804 is that if the Twelfth Amendment is not
enacted, the Federalists, hopelessly outnumbered in the Electoral College,
may try (despite warnings by Hamilton) to make Burr President--this time
in the Electoral College rather than in the House. It is possible but by
no means certain that the Republicans will be alert to the danger, and
will arrange for enough Republican electors to vote against Burr to offset
any Federalist votes in his favor.
(4) No Hamilton-Burr duel (at least as we know it) so Hamilton presumably
remains alive.
(5) No Burr conspiracy (whatever exactly it was...) and hence no trial
(with that trial's lasting contributions to American jurisprudence on the
law of treason, Presidential immunity, self-incrimination, etc.).
Any thoughts?
[1] This change was controversial, as is shown by its passing the House of
Delegates by only four votes, despite Republicans outnumbering Federalists
by two to one in that body. Suppose it had failed? According to John H.
Broussard, *The Southern Federalists 1800-1816* (p. 30) "The General
Ticket Law probably did keep some Federalists at home, but even if
electors had run by districts, Adams would have had no more than two of
Virginia's twenty-one votes." So Jefferson and Burr would still have
gotten more electoral votes nationwide than Adams and Pinckney.