There were some arrowheads made specifically to fold burning embers or stuff like that, but the majority of arrows, as shown by the movies, would be put out simply by doing their job. Plus, it makes something more visible to the enemy, and therefore more easily dodged or avoided. The most egregious example is probably Timeline, when the British commander tells his archers to fire “night arrows” after many volleys of brightly-burning flaming ones. These end up being just arrows with nothing special on them.