This technique is not possible with Bowser or Giga Bowser in the NTSC 1.2 and PAL versions, and is as a result not legal in most tournaments. This technique returned in Project M for Bowser, though it was removed in version 3.6.
How to perform
Due to the frame conversion between an aerial and grounded Fire Breath, the game skips the startup and goes straight to the attacking animation of Fire Breath. Effectively, Fire Breath will become a frame 2 option if done optimally.
Thanks to this broad requirement, landing with Fire Breath at any point prior to the attack coming out will cause a flame cancel. The nature of the tech gives it inconsistency in how many frames are saved; though this also means landing as close to the ground as possible gives the best chance of an optimal flame cancel.
With this technique, Bowser can cancel the startup lag of Fire Breath if he lands during its initial frames.
Since landing with Fire Breath in general will cause a Flame cancel to occur, there are many opportunities to utilize flame canceling. Through Flame canceling, it's possible to get some unique mindgames with Bowser, as being in the air in neutral inherently carries the risk of a flame cancel for the opponent. It's also possible to initiate it through dropping through platforms, or it could be used during tech situations with 1 frame to act. It also makes Fire Breath a slightly less situational edgeguard tool. It's also possible to jump from the ledge and initiate the tech. In this context, Bowser can get out of some difficult ledgetrap situations, which is vital given his large hurtbox. Thus, he doesn't have to always rely on his great edge attack.
While it is possible to SDI towards Bowser and exploit the minimal hitstun to punish him in the endlag, the lack of time to react can arguably lower the risk of this, allowing Fire Breath's tendency to be punishable on hit less of a problem. However, this doesn't mean it isn't, which means against experienced players makes it extremely risky to pull off. As a result, whether the tech actually salvages Fire Breath is debatable.
It is also possible for Giga Bowser to flame cancel, though this was also patched out in NTSC 1.2. Giga Bowser arguably uses it better due to his very low short hop and shorter jumpsquat, allowing for much quicker access than Bowser. If a player uses Giga Bowser through hacking, it is possible to set up edgeguards with Fire Breath quite quickly, as it goes quite far off-stage. As a result, it is possible to KO opponents very efficiently through the use of this tech in conjunction with his large hitboxes (most notably his down tilt, which goes exceptionally far off-stage).
Strangely, Flame cancelling remains possible with Kirby throughout all versions of Melee, likely an oversight by the developers. Due to his air speed and high short hop however, it is difficult to make effective use of the tech; it tends to be easier just to use it normally. However, it can be used during tomahawking situations or with a turnaround special as a mixup option in the matchup. It can also be a useful landing option, which is important when considering Kirby's vulnerability to juggling. It can be used as a ledge option as well, much like with Bowser.