As I understand it in laymen's terms, an air stone or any openly flowing water (like a filter overflow) and surface tension breakage allow CO2 to exchange with air, thus reducing the carbon dioxide and replacing the oxygen. Without the exchange, even fish will die rather quickly of suffocation.
Between my skimmer, hang-on filter and cannister filter (all in my sump), I have removed the air stone from Octane's tank but would loose all my clowns in a matter of hours in my tall skinny tank that is filtered only by a cannister. Since there is very little surface area and no openly flowing water, I have learned, the hard way, that a battery backup designed to cut ON if the power goes off is an absolute necessity for that tank.