Maybe there should be a category for wishes rather than ideas since there is no technology here, just a desire. The desire is good but it's not a solution.

There is a tremendous amount of energy created in a forest fire. Also the fire develops a huge amount of CO2. If the energy of the fire could be utilized to redirect the created CO2 downwards onto the fire, it would smother itself. Perhaps huge helicopter type fans sustained above the fire by the fire's energy could force the flow of gases back downwards and prevent fresh oxygen being sucked into the updraft from the sides of the fire. Sounds like a very expensive operation but it is a new approach. Perhaps impractical.

The idea of using the fire's own energy to help drive oxygen away from the flames intrigued me, and I thought for a while before a lightbulb came on...

The easiest way to use a heat source as a source of work is a heat engine, which applies heat to a volatile material to induce a phase change and thus pressure change which performs work. So what you require here is a volatile material which will absorb some of the fire's heat energy (cooling it in the process), and in doing so expand thousands of times to form a blanket of nonflammable, non-oxidising gas which separates the fuel from oxygen. As it probably won't be practicable to re-condense it, the material would need to be cheap and widely available in large quantities. A suitable material might be ... water.