If creatures targeted by Living Inferno's ability all die, what happens?
The ability fizzles if and only if,  all of it's targets are gone, so long at least one of it's targets is still there, It will still resolve and deal damage to that target, but if they're all gone, the ability is countered(fizzles)
Creatures on the battle field can't perform actions(such as dealing damage) if they are illegal targets of the ability, and they're illegal targets of the ability if they've changed zones(such as from the battlefield to the graveyard).
If they died while  living Inferno's ability was on the stack, than those creates would not deal damage.


both of these points are covered by section 608.2b of the comp rules


  608.2b
  If the spell or ability specifies targets, it checks whether the targets are
  still legal. A target that’s no longer in the zone it
  was in when it was targeted is illegal. Other changes to the game
  state may cause a target to no longer be legal; for example, its
  characteristics may have changed or an effect may have changed the
  text of the spell. If the source of an ability has left the zone it
  was in, its last known information is used during this process. The
  spell or ability is countered if all its targets, for every instance
  of the word “target,” are now illegal. If the spell or ability is not
  countered, it will resolve normally. However, if any of its targets
  are illegal, the part of the spell or ability’s effect for which it is
  an illegal target can’t perform any actions on that target, make
  another object or player perform any actions on that target, or make
  that target perform any actions. The effect may still determine
  information about illegal targets, though, and other parts of the
  effect for which those targets are not illegal may still affect them.
  
  Example:
  Sorin’s Thirst is a black instant that reads, “Sorin’s Thirst
  deals 2 damage to target creature and you gain 2 life.” If the
  creature isn’t a legal target during the resolution of Sorin’s Thirst
  (say, if the creature has gained protection from black or left the
  battlefield), then Sorin’s Thirst is countered. Its controller doesn’t
  gain any life. 
  
  Example:
  Plague Spores reads, “Destroy target nonblack
  creature and target land. They can’t be regenerated.” Suppose the same
  animated land is chosen both as the nonblack creature and as the land,
  and the color of the creature land is changed to black before Plague
  Spores resolves. Plagues Spores isn’t countered because the black
  creature land is still a legal target for the “target land” part of
  the spell. The “destroy target nonblack creature” part of the spell
  won’t affect that permanent, but the “destroy target land” part of the
  spell will still destroy it. It can’t be regenerated.


This is also supported by the Gatherer ruling for pit fight


  1/24/2013     If either target of Pit Fight is an illegal target when the
  ability tries to resolve, neither creature will deal or be dealt
  damage.


take note that  pit fight does not fizzle in this case, because it still will have 1 target.



if  pit fight isn't good enough for you because it uses the fight mechanic, than the ruling for Flesh/Blood also applies


  4/15/2013     As Blood is resolving, if only one of the targets is legal,
  Blood will still resolve but will have no effect: If the first target
  is illegal, it can’t deal damage to anything. If the second target
  creature or player is illegal, it can’t be dealt damage.