Does Heatran GX's "Burning Road" Ability work if it becomes your Active Pokemon as a result of an attack like Archeops' "U-turn"?
As long as Heatran GX becomes your Active Pokemon during YOUR turn, it can use Burning Road.
Source: Unified Minds FAQ (2019-07-19), TPCi Rules Team (2019-08-01)
Can I use Arcanine's "Burning Road" Ability any time during my turn, or do I have to use it the moment Arcanine becomes Active?
As soon as Arcanine goes from the Bench to become Active you must choose immediately whether you're going to use Burning Road or not; you cannot delay the action until later in the turn. If you choose not to use Burning Road at that time, you could use it later if Arcanine somehow goes back to the Bench and becomes Active again. Also, if you use an attack like Growlithe's "Hind Kick" and Arcanine becomes Active, you could use Burning Road if you haven't already used it that turn.
Source: Evolutions FAQ (2016-11-03), TPCi Rules Team (2016-11-03)