Is Fluorine a conductor of heat?

No, fluorine is not a good thermal conductor, in fact, like most gasses it is really bad at it.

Thermal conductors are rated by W/cmK. Without going into the explanation, a higher number represents a better conductor, and 0.0 means no conductivity at all. Fluorine is rated at 0.000279 W/cmK. By comparison, copper, which is an Excellent conductor, is rated at 4.01 W/cmK.