What happens when I target an indestructible card with an "if that would die this turn, exile it instead" effect?
No. 

Lethal damage will not cause an indestructible creature to die.


  700.4. The term dies means "is put into a graveyard from the battlefield."
  
  702.12b A permanent with indestructible can’t be destroyed. Such permanents aren’t destroyed by lethal damage, and they ignore the
  state-based action that checks for lethal damage (see rule 704.5g).
  
  704.5g If a creature has toughness greater than 0, and the total damage marked on it is greater than or equal to its toughness, that
  creature has been dealt lethal damage and is destroyed. Regeneration
  can replace this event.


However, if another effect causes the indestructible creature to die (for example through sacrificing), Annihilating Fire's effect will cause the card to be exiled.

A ruling on gatherer:


  10/1/2012     A creature dealt damage by Annihilating Fire will be exiled
  no matter why the creature would die. It could be destroyed by another
  spell or ability.