I don't believe that flaming arrows were used with humans as their primary target, and as such wounds caused by them might not be subject to the sort of documentation that would survive to the present day. I believe that flaming arrows were typically intended to set fires, such as in a siege, and not so widely used in open battles as a typical missile. Consider the following passage by Julius Caesar from The Gallic Wars, Book 5, Chapter 44:

On the seventh day of the attack, a very high wind having sprung up, they began to discharge by their slings hot balls made of burned or hardened clay, and heated javelins, upon the huts, which, after the Gallic custom, were thatched with straw. These quickly took fire, and by the violence of the wind, scattered their flames in every part of the camp.

The Gallic Wars has a lot more to say about the use of fire in warfare, but from skimming, it seems to always relate to trying to actually set a fire.

With that all said, I'll be watching in case an expert does know more!

Try /r/askscience