As long as you can continue to "suppress it momentarily" long enough for the material to cool below its ignition point, you've solved the problem.
To be honest, I'm sort of skeptical, especially of the "glorified speaker" approach, but as it happens there are ways to produce some very loud noises at 30-60 Hz which is very well-established off-the-shelf tech: http://epb.apogee.net/res/rehpuls.asp This used to be a standard furnace technology in American homes. Much like cars, it would be tuned to be as quiet as possible (which I've heard could still be annoyingly loud), but just as I'm sure you've all heard a car without its muffler and marveled at the difference, I'm sure something based around this technology, only meant to be as loud as possible, could achieve some serious decibalage, quite efficiently. Probably a great deal more efficiently than converting something to electricity and then converting that to sound.
My understanding is that it doesn't starve the fuel of oxygen, but rather starves it of activation energy. I'm fuzzy on the chemistry, but I remember it being like this:
fuel + O2 + activation energy -> CO2 + H2O + energy
The propagating sound waves move two of the input components of the reaction.
fuel: stationary
O2: moves, but is replaced by more O2
activation energy: moves and is missing from future reactions unless/until new activation energy is introduced.
Again, I took chem long ago, but I think that's what's going on.
Or you could equipate an electric ashtray with this.