^ Since the impure king doesn't possess people, they were free to go ahead and hurt the host as well - and without a host the demon has to go back to Gehenna. So, in a way you are right that it'd hurt the demon, but it's more directed towards the host (a "material'' of Assiah) than the demon itself.
But black flames harm that of which is connected to Gehenna, so it's the one actually doing the harm in a demon possession vs. red flames, which just burns the host.
Considering what chino said about the kanji and such - it sure makes Gehenna much more of a spiritual realm to me than a whole another world - I'm pretty sure if you took red flames to Gehenna they wouldn't have much affect on the demons.
Unnlleeesssss
Humans souls are in Gehenna = vulnerable to black flames
Demon souls are in Assiah = vulnerable to red flames.
Personally, I don't think this is so (Mind, this is not talking about how they need to hide their hearts and what-not; since Mephisto states if the heart is crushed they'd surely die, but it wasn't mentioned if a soul was damaged that it would kill the person. So, I think the hearts and souls are different.)