As it rises it is replaced with fresh air with a supply of oxygen to sustain the flame. The result is that the oxygen in the vicinity of the flame is rapidly depleted and the flame dies due to oxygen starvation. Now that we have two opposing views it is time for peer review. Greg, I believe your assertion that similar objects of different masses fall at the same rates is true in a vacuum, but clearly a candle could not burn in a vacuum. I am with Dr. Blackburn's beautiful reply.
Fires clearly depend on the flow of combustion reactants into and products out of the reaction zone.
Of the varying terminal velocities of candle, air, and elevator in atmosphere only the last is relevant, since all objects are enclosed in the elevator. You could try the experiment in the Space Shuttle, but I would not think they would allow open flames. When not in a vacuum, i.e., falling through the air, there are viscous effects which affect the rates. However, this problem specifies a closed container so the viscous effects act only on the container. Argonne National Laboratory, Division of Educational Programs, Harold Myron, Ph.D., Division Director.