[Sombre]

I'm pretty sure that,...

Etherealness negates your ability to hit a creature. Since the fire damage from flaming arrows/weapons is only dealt if the arrow/weapon hits in the first place, it is safe to assume that either both normal damage and fire damage land, or neither do.

I would guess that being immune to fire and ethereal doesn't change the chances of fire weapons hitting the monster - though the secondary fire damage wouldn't be dealt.