I couldn't find a specific place this is covered by the rules, but if you have a burning section of your capital and an effect which can heal damage from your capital, can it "un-burn" the section?

No. Once it's burning, damage is removed from it so there's no damage to heal. You can't assign more damage to it either.

But you can still attack one.

Yes, you can, which can be relevant for triggering scouts abilities, Sack Tor Aendris, etc.

As an aside, the most common mistake I see beginning players making is assuming that a burning zone is effectively not there any more (i.e. the units, supports there don't do anything and they can't play more into the zone). I'm guessing this is why they addressed that specifically in the official FAQ.