Why doesn't the carbon dioxide produced by a fire put out said fire?
The fuel for the chemical reaction is the CH4 (in this case) and the oxygen.  Assuming the CH4 isn't going to run out any time soon, the chemical reaction will continue as long as there is enough oxygen accessible. The force behind fire extinguishers can replace the oxygen available to the fire in its immediate area with CO2, and without one of the reactants (oxygen) available, the reaction won't continue.

The CO2 produced by the fire in an open area isn't going to replace the oxygen of the surrounding area.  However, in a closed system (for example, if you put a lid on top of a burning candle), then the air inside of the candle will become deficient in oxygen and rich in CO2 after a certain amount of time and the flame will go out, even without blowing on it or forcing it out.