The first showing of Fire Breath (ファイヤーブレス) is halfway through Act 1 of Angel Island Zone, where it burns up the entire level. At the end, you actually fight the machine. It has a flamethrower and bomb launchers, though it only uses the latter if you are playing as Knuckles. If you have the Fire Shield, just go below it and deck it. If not, wait until it comes down, then hit it a few times. Watch out for when it shoots a flame when it comes down. After six hits, the sub-boss is destroyed.
When Knuckles fights the machine, he does it underground, and the boss has a new attack. It will fire three missiles after shooting the flame, so dodge those as well.