@Sylas Vallemental also has Disadvantage on the Slam (which I presume is against S2).
Because they are "on fire" does not negate the fact that you cannot see them, so you still have Disadvantage (the faerie fire spell negates the power of invisibility, not normal fire).
This is two-pronged:
A) there is no rule for "on fire", and I'm inclined to rule that the creature is not completely covered in fire, e.g. a humanoid-shaped inferno. The "on fire" creature will have a much harder time trying to hide, but they are not rendered visible.
B) the creature in question is an air elemental...I think I'm giving leeway by allowing air to be "on fire" to begin with. As there is not a game mechanic or condition for being "on fire", thus stat blocks do not provide immunities for it. So using "common sense logic" to be on fire, something has to be flammable...perhaps air elementals are carbon dioxide or helium.
So, I'm ruling they are "on fire" and take the extra fire damage, but you still cannot see them well enough to negate the disadvantage from being invisible - just a mote of fire swirling in a chaotic fashion about the air.