Just to prove how nerdy I am:
Fire is a series of actions and changes that produces a result — a process. In fact, it is an oxidation process (called combustion or burning) that gives out heat and light energy as well as glowing gas and a small amount of plasma.
So, talking about fire's mass is like talking about the mass of digestion, or boiling, or getting a driver's license although it certainly makes us think about what fire is. But a process doesn't have mass.
Flames are another matter. They are burning gases and gasses certainly do have mass.