Does Deflecting Palm cause Furnace of Rath to activate twice?
Your opponent will lose, providing you choose the correct order of replacement and prevention effects.


Your opponent casts Lightning Bolt.
You respond with Deflecting Palm and let it resolve.
You now have 2 replacement / prevention effects attempting to modify Lightning Bolt.
By rule 616.1



  616.1. If two or more replacement and/or prevention effects are attempting to modify the way an event affects an object or player, the
  affected object’s controller (or its owner if it has no controller) or
  the affected player chooses one to apply, following the steps listed
  below. If two or more players have to make these choices at the same
  time, choices are made in APNAP order (see rule 101.4).



Since you are the affected player, you choose the order of replacement.


Option 1 - Apply Furnace's Effect First


This doubles the damage of Lightning Bolt from 3 to 6.
As per 616.1e, you now repeat the process with any applicable replacement effects



  616.1e Once the chosen effect has been applied, this process is repeated (taking into account only replacement or prevention effects
  that would now be applicable) until there are no more left to apply.



The only applicable effect left is Deflecting Palm.
Now your opponent is affected by Deflecting Palm dealing 6 damage to him / her, so they would make any applicable choices for replacement effects.
The only replacement effect here is Furnace of Rath again, which doubles the damage from 6 to 12.


Option 2 - Apply Deflecting Palm First


Lightning Bolt's 3 damage is prevented.  Deflecting Palm now deals 3 damage to your opponent.  Note that at this point, Furnace of Rath's replacement effect no longer is applicable to the Lightning Bolt event, so we don't worry about it anymore.
Like above, Furnace of Rath's effect applies to Deflecting Palm's damage, doubling it to 6.


So if you do it right, your opponent loses.