After doing some more reading I found most candle flames aren't hot enough to be conductive. In order to make the charge carriers (electrons) mobile enough to conduct current, the flame has to be hot enough to give the electrons the energy needed to break free from their atoms. With some experimenting I found that some metals work well and others will not, when exposed to flame copper forms a skin of copper oxide which will insulated it from the flame quite a bit, clean steel seemed to work a lot better. A little more tinkering showed that geometry is important as well, getting the most metal possible in contact with the flame increased the conductivity to the point where a very weak audio signal could be passed, and I could make this video.
With a pair of large parallel plates, and a very hot flame it could even be possible to generate current using a magnetohydrodynamic generator! Putting strong magnets perpendicular to the plates and the flow of the flame would cause the charges in the flame to migrate to one plate or the other, resulting in a measurable voltage.