Dust Tornado is activated, presumably targetting the face-down Royal Decree. That Royal Decree is chained to it.
First, Royal Decree resolves, and from then on all Trap effects on the field are negated (except itself, obviously). Then Dust Tornado tries to resolve, but it can't (because of Royal Decree's continuous effect). Dust Tornado's effect is negated, no cards are destroyed by it, and it's sent to the Graveyard by a game mechanic (it's been used).
Royal Decree survives, Dust Tornado ends up in the Graveyard. Furthermore, the second effect of Dust Tornado cannot be used either (to Set a Spell or Trap Card of your own).