As the subject says, can a burning skeletal champion suppress his fiery aura?
You know, so he can go inside wooden buildings without burning them to ground or carry non-metal objects, or sneak around at night.
The Fiery Aura of a Burning Skeleton doesn't say it can be turned on/off, and no action is listed to activate/deactivate it. So I would say, at least by the rules, that it can not be deactivated. Its always on.
Fiery Aura (Ex): Creatures adjacent to a burning skeleton take 1d6 points of fire damage at the start of their turn. Anyone striking a burning skeleton with an unarmed strike or natural attack takes 1d6 points of fire damage.
Also, it specifies creatures take damage. No mention of objects in the area taking damage as well. It would make sense that objects are harmed, but by a strict reading of the ability, only creatures ever take damage from it. (I would never use it that way, but its at least possible it works that way because of how it is worded.)