No. The rules are somewhat, but that is not one of the possibilities.
An election would definitely still happen.
Some of the possibilities are:
a) The ballot would still show Trump's name and Biden's name; the electors in the states that Biden "won" would pick whom to give their states votes in the electoral college. The result of this would probably be that if Trump didn't win a majority of the electoral college, then no one would, and the House of Representatives would pick from the top four finishers (this last happened in 1824, although under different circumstances).
b) The ballot would still show Trump's name and Biden's name; the electors in the states that Biden "won" would give their states' electoral college to Biden's running mate (this has never happened).
c) The Democratic party leadership would pick someone else to have listed on the ballot instead of Biden.
d) Similar to (a), except that the House of Representatives can't agree on who should be President, so no one is elected President, and whomever has been elected Vice President acts as President for the next four years.
In any event, I think the two most likely outcomes are:
a) Republicans win the electoral college, Trump gets a second term, or
b) Democrats win the electoral college, Biden's running mate (Vice Presidential candidate) becomes President or Acting President.