FYI Gila BLM district lifted fire restrictions July 30.
https://www.blm.gov/programs/public-safety-and-fire/fire-and-aviation/regional-info/arizona/fire-restrictions
Phoenix BLM district still has restrictions.
District map https://www.blm.gov/sites/blm.gov/files/Arizona%20Grazing%20Map.pdf
Also Coconino National Forest is under stage 1 fire restrictions which unique to them I do not believe prevents discharge of firearms for recreational purpose. But you should look more in to that for your self.
Regardless of where you go, be very careful as way to much of my favorite parts of this state have burned this year. Unfortunately target shooting does start a lot of fires. Try a night shoot and see the sparks that happen down range, especially with rifle velocity projectiles. Best to have a soft dirt backstop with no burnable vegitation in the vicinity.