If TBB edits his answer before I finish writing this, then I'll hide it, but in any case:
Flash Fire does not redirect Fire attacks. Flash Fire only activates when the Fire type move (including HP fire) is directed at the Pokemon with the ability, i.e. Chandelure.
Lightningrod and Storm Drain are the only abilities I can think of that attracts all the attacks of a single type (electric/ water respectively), though with some limitations (Judgement electric/ water is not redirected).
Motor Drive, Sap Sipper and Water Absorb will not redirect attacks, but only negate the effect of electric/ grass/ water moves on the Pokemon they're directed to.
Compare: Flash Fire as opposed to Storm Drain or Lightningrod.