You may be thinking about closures due to fire danger. Not sure if they are going to close the trails, but I was there this past weekend and things were wet due to the almost daily rains, so my guess is that there are not going to be any closures. Its even possible that existing camp fire restrictions may have been or will be lifted.
The trails on Mt. Elden were very nice - no dust, packed. Mt. Elden is the must do place to ride in Flagstaff. I suggest top of Sunset to Little Bear to Shultz Creek.