I get tons of axetinguisher kills. The main trick is to hit them with a small burst of flame, before they even really notice, and then switch to the axe and finish them before they can respond. One amusing thing to do is to hit them with the poofer right after the ignite, then instantly switch to the axe, before the whole poofer animation even goes off. It's usually disorienting enough that they have even less chance to respond.
I was reading all these Axetinguisher questions and I was gonna suggest exactly this, but it looks like Bal beat me to it.
I don't really understand why Brent is suggesting it's a good pairing with the flaregun since the whole point of the flaregun is to ignite people at a distance where as the axetinguisher is a melee weapon... So why not just ignite them with your flamethrower?
Good use of the Backburner, by which I mean ambushing people and never attacking them from the front or at least, quickly getting to their back when you can makes the axetinguisher a rarely necessary weapon. If you are getting back crits or think you are about to, switching to it rather than just letting the damage boosted flames do their job is really just a showboat move that can only hurt you since you have to waste a valuable second or so switching to the axe and then actually manage to hit the guy with it.
This said, it does do more damage in less time and it does do so from any angle so when you run into a situation where you just can't get to the back of someone, they are already in melee range and on fire, it can score you a nice kill you wouldn't have gotten otherwise.
But Bal's example is where it can really shine in the hands of an experienced Arsonist. Poofing someone while aiming the poofer at a slightly upward angle will cause them to fly into the air. This can be very disorientating to many of the traditionally anti-pyro classes like Soldiers.
The best way to use this is in areas that are fairly closed in ground area wise, but that have high ceilings. The trick is to poofer someone such that they get jarringly launched into the air towards a near by wall. The wall prevents them from getting pushed too far away from you, allowing you to quickly close any horizontal distance between you and meet them as they land with a nice axetinguishing axe hit. You can normally one shot just about any class by doing this except for the obvious case of a healthy Overweight Weapons Guy.
As brutal as that trick is if you can pull it off reliably, ambushing people with the backburner while not always possible, is a much more reliable way to be almost as deadly. So I only really suggest Poofer + Axetinguisher play if you think it'll be fun and don't mind having to practice it for a bit to get it down. It can be a lot of fun whenever you get bored of "standard" backburner play. Just keep in mind though, using this strategy you will be at a distinct disadvantage against other pyros who have the back burner as your axetinguisher is useless against them while their back crits can still destroy you. When faced with this sort of duel situation, I find I have the best luck by attempting to poofer them out of flamethrower range and then just blasting them with my shotgun while I run away.