It's in the Fuego Ironworks. To get there, go to Route 205, next to the Valley Windworks, and surf west up the river and north from Floaroma Town. The Ironworks features a lot of spin pads, most famously used in Team Rocket buildings, and the TM near to Mr. Fuego, who will trade a Shard for a Star Piece.
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You only can learn flamethrower if you have a fire-type pokemon. Look for TM35.
Houndoom learns Flamethrower at lvl 48.
you can get it by going to fuego fireworks but you nedd surf monkey
In Platinum version Houndoom learns Flamethrower at level 48. in diamond and pearl Houndoom also learns it at level 48.
Veilstone Game Corner Exchange, the TM (35) is bought for 10,000 coins which is about 200,000 dollars