Flaming arrows are great against siege towers at close range. Wait until the siege tower is at the wall or close to it so the archers can fire at the sides. Within a few volleys the tower will probably catch alight. And if the tower has been at the wall for a while, when it burns down, you'll probably kill nearly the entire unit assigned to the tower at once. Keep some units between the archers and those coming up the tower though to give them time to burn it down.
Flaming arrows are great against elephants too. Gets them to rout or run amok faster.
I sometimes switch to flaming arrows when I want to be able to preserve arrows but still keep up a constant barriage of arrows. Kill rate might not be as good, but at least the archers won't run out of ammo so quickly, plus the enemy suffer from morale losses for a longer time.
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