I think the flame should go out IMMEDIATELY if you un-equip the weapon. Or add a water canteen to the trader guys in the dropship that you can buy and carry on your belt so you can put the fire out. Maybe it can also be used to dilute poisons on the ground so they disappear a little faster. Dunno.
It sucks though trying to defend the base on Koth with a flamethrower, lol. They manage to hit me with a bullet and then I'm on fire, and they keep shooting as I'm panicing trying to unequip the flamethrower thinking it'll put the fire out quicker
..