I expected to read very detailed answers, there are literaly tons of reasons to not use flamethrowers, Just glad most reasons was not already told in other answers.
Flamethrowers are weapons designed to kill people finding refuge in very small spaces (American invasion to Japan Isles) like bunkers.
First of all it would not be a very effective weapon against zombies (not if you are in immediate danger):
- They do breath? Mostly not (they do smell, but in most movies Zombies just walk underwater, that made me think they do not breath)
- They do rely much on senses? Not likely, and could still wander even if blinded
How much time is required by a flamethrower to kill a zombie? The time to severly damage the brain or at least to burn muscles that allow moving it. How much time is required to cook meat? More than few seconds for sure.
You don't want a weapon that require 1 minute to kill an enemy that can be killed by 1 bullet
Not to mention that 1 gun with 1 bullet is much more lightweight than a full flamethrower suit.
People dying because of flamethrowers mostly dies because of damages to plumbs or because start running as crazy under enemy fire, or just because commit suicide with weapons at its disposal (or because explosives it was bringin are ignited by the flames).
That weapon seems much more usefull to kill humans rather than zombies.
- You cannot say if someone killed by a flamethrower was a Zombie or a Human (may matter to know for public health stuff)
- You cannot recognize who was the burnt person (easy kill someone and then disguise by removing clothes etc)
- Using flamethrower inside buildings is very Dangerous
- You could kill anyone who was hiding from zombies (you burn zombies in a house where there was children hiding)
- Fuel is a rare resource in a zombie apocalypse world, you usually prefer it for vehicles. Assuming flamethrowers use fuel
- Difficult to create ammunitions: gunpowder much easy to create, you just need to know which ingredients to mix, instead fuel and flamethrowers ammuntions requires a full oil industry (in addition to weapon industry).
Another important reason:
What do zombies eat? Meat. What do you think zombies would do if they start to smell that delicious barbecue of burning meat? Zombies would just come in hordes to your current place.