Where to get flamethrower on Pokemon platinum?

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.