Well, the Electoral College have already voted (in December 1792) so Adams is unquerstionably Vice-President Elect. If there is no President on March 4, he presumably succeeds to the position, and continues for the next four years. He has, of course, already been POTUS for the unexpired portion of Washington's first term.
If he makes himself as unpopular as he did in his OTL term, then he loses to Jefferson in 1796. Biggest difference is that this, in all probablility, means that John Marshall does not become Chief Justice. Any thoughts on who Jefferson might appoint, and with what results?
Also, the tie between Jefferson and Burr (assuming Burr is still the running-mate) is less likely, as in 1796 the Electors' second votes were far more scattered than in 1800. So he stays in good standing with the DRs, and may be nominated in 1904 to succeed Jefferson.