No, as it stands right now a spell doing fire, wind, ice, unaspected, lightning or earth damage is just for flavor text. Even the Bard's Requiem song, which boosts "elemental" damage, benefits all magic from a Black Mage or Summoner, even though nearly every summoner spell is unaspected. Likewise a Bard's own Windbite, which does Wind elemental damage, does not benefit from Requiem. If this seems silly and poorly-worded, that's because it is.

You can blast Ifrit with Fire and it'll hurt him just as bad as blasting any other enemy would. Many of us are still hoping they decide to do something interesting with elements in the future, but the last official word on the matter was a general "no."