If Burnham wins there will be a leadership fight within Labour. In all likelihood Starmer will lose, but it may be a bloody fight. The new PM does not have to call for a new national election, but the situation may be such that he will feel forced to do it.
If Burnham loses the by-election there will not be any challenge to Starmer, or a very weak one that he will be able to win, in which case he will sit as PM until his five years are up.
If Burnham wins there will be a leadership fight within Labour.
There won’t be a fight, Starmer will stand down for Burnham, without a fight.