What happens when the immolator's burning brand misses (6-)? The GM gets to make a move, sure, but does the immolator get their weapon, even if it's without the extra tags (+1 or not dangerous, etc)? And without such a weapon, can the immolator deal damage?
It's entirely up to the GM whether or not a weapon appears, and what tags it does, or does not, have.
Remember from the basic rules:
Most moves won’t say what happens on a 6-, that’s up to the GM but you also always mark XP.
That means your GM might say:
A massive, flaming weapon appears. It's +1, messy, dangerous and it appears to be unstable and drawing in fire from around it, growing more powerful with each passing second. Also, the entire Inn is now on fire. Mark XP. Fightgar, your buddy just summoned a flaming weapon and now the entire room is on fire; what do you do?
And if so, you did summon a sword (but you're not happy about it).
He might also say
You draw in the flame, but nothing seems to happen. The Goblins around you point and laugh, before closing the circle around you, boxing you in. Mark XP
And then you don't have a sword. (and you're not happy about it)
The only thing you know for sure, is that you're not going to enjoy what is about to happen; it's up to the GM how that manifests.
As for dealing damage without a weapon; anyone can deal damage without a weapon. Your damage die (which is a d8 for an Immolator) is always available. Whenever you deal damage in the fiction, that's how much damage you do. It does not matter if you use your Burning Brand, a weapon you bought/stole/found, a rock you picked up from the ground, or your bare fists: if you deal damage to something in fiction, you roll a d8.
(The specific weapon you use does decide whether or not you actually deal damage; a punch is unlikely to damage a Dragon, but your Burning Brand probably will. But if you fictionally stab the Dragon in the eyeball with your thumb, that'll probably deal a d8 as well.)
I think the answer to this question is pretty simply stated in the Move:
The weapon always begins with the fiery, touch, dangerous, and 3 uses tags.
So on a 6- the weapon still always begins with those tags.