There are no rules regarding how good you have to play. It's almost impossible to measure or control such things.
But people tend not to like you if you do something like that.
To answer on your scenario, it's impossible. Depends on who's playing who, are they in relegation zone? Can they qualify for Europe? Do they play the FA Cup final next week( then you give some key players rest if there's nothing else to gain from the game). Are they rivals? Is the other League winner their rival.
Can't find it at the moment, but their was an instance Marseille supporters demanded of their team to lose because otherwise their rivals would become champion, I think it was for the Marseille-Montpellier game in 2012.
And, it's not that uncommon for teams who have nothing to gain anymore let the other team be. There are a few famous examples. In Dutch there is a word for it, salonremise. Litteraly it means draw, but it's used for not only draws.
The most notorious is the Disgrace of Gijón. It was even after this game FIFA put in place a rule so that the third game of the group stage have to be played simultaneously. West Germany needed a win to continue, and Austria would progress to if they only lost 1-0 or 2-0. West-Germany scored quickly and there after they just passed the ball along. Algeria was the victim of this disgrace.