i'm not too sure on how well a sling burns, but my thickly braided rockman's can take a decent amount of fire with a little amount of actual damage. while playing around etc burning off hairs from my sling i learned to hang my slinging on a hook, or for some extra fun on your finger with the loop at the top. light the bottom of the sling (should be where a whip cracker is) and the whole thing catches on fire and dies out after a second if it's a really hairy sling. the fire lights at the bottom, and quickly rises up. if it's really hairy and you really get it lighted it looks like the whole sling is burning, if it isn't a sisal yeti monster then the flame will quickly rise up. no damage at all too the cords.
after seeing that i decided to do even stupider stuff. when you burn it this way, the pouch slightly catches around the edges (rockman, thick straps), but not the actual pouch itself. so of course i needed to do something potentially dangerous to remove the hairs on the pouch. well i couldn't figure anything out
so i just did it normally, light the area, wait for it to burn. unfortunately, the pouch was pretty hairy and there wasn't enough air to set it on fire correctly. so i had to hold the lighter there for a few seconds before it burned right. no damage at all
long story (with many grammar mistakes) short, after a few seconds of intentional burning and no protective equipment the sling took no damage at all, minus a few hairs. so i doubt any serious methods would be needed to sling flaming ammo or hot ammo once or twice. if the sling isn't allowed to cool off after each toss then i guess some adjustments would be needed.
i don't think submerging the sling in water is a good idea. if the slings were wool then the sling might shrink or felt after some long use. if the cord has glue/resin in it then it will become hard and stiff in water (at least thats what i think makes a sling using modern materials stiff when dunked in water). i don't think a metal sling is viable either, it's much too heavy, expensive and possibly dangerous. i wouldn't want to be hit by the release cord of the sling.
a jax/thomas captive release caulk tube design using ancient materials sounds like a viable method.
if i had to do this i would like someone to load for me so that i can immediantly sling it when i drop the ammo.