They are flying too close to the sun... probably the villagers are going to do something bad, demon queen will defend the hero and the hero will know about demon queen´s identity... or maybe the hero will see other demons doing horrible things...
No, demon king, you don't burn the letter. You steal it, so that you know who the hero's allies are. Hero can always write a new letter, but you wouldn't get a chance to intercept his mail every day.
You don't have to actually go yandere to the hero's allies, by the way. There are more subtle ways to do it. Like in that manga where the Hero can't depart on his journey because he's too busy with all those lewd girls.
He doesn't even realize that he has already defeated the demon king from the beginning. The war was over before it started when the innocent hero turned the demon king into a hopeless shotacon.