Does fire create air resistance? Does fire create air resistance/drag?  So, for example, would it be harder to swing a flaming sword than a normal one?
 A: No. Fire doesn't create resistance. Resistance is offered by drag force because we're pushing the medium (which pushes back on us) which obstructs our path. It depends on the density of the fluid, our velocity (for high Reynolds numbers), the area of the substance which obstructs the fluid flow. This can be seen in our theory $F=\frac{1}{2}\rho A v^2C_d$
A sudden definition for fire (as seen in everyday life) can be the burning of something by rapid oxidation. In other words, you're losing the mass of the material which catches fire because the it's slowly (when viewed in normal scales) stolen away by oxygen which leaves the molecules into some kinda excitation. You're lucky to see the fireworks which is actually due to the emission of blackbody radiation from the flying away molecules.
To be precise,, Fire actually helps reducing drag. After swinging for some time you'll notice that it'll be easier for you to swing the sword, because the it has been dissolving in air the whole time (i.e) the area of contact with the fluid flow $A$ is getting reduced (though a negligible effect). Also, the density of the fluid in front (shielding the whole object) is quite less than the density of fluid $(\rho_{air}>\rho_{fired})$, leading to easier motion of the sword...
A: I think it is easier to swing the fire sword than the normal sword. 
Take the (subsonic) equation for air drag $F_D$: 
$$
F_D = \frac12 \rho_{\mathrm{air}} C_DV^2 S
$$
where $C_D$ is the sword's drag coefficient, $S$ the frontal area, $V$ the speed at which you swing it, and $\rho_{\mathrm{air}}$ the air density. Take also the ideal gas law, 
$$
\frac{P}{\rho}= RT
$$
with $P$ pressure, $\rho$ the specific density, $R$ the universal gas constant and $T$ the temperature. 
From the ideal gas law, it is obvious that an increase in temperature would result in a decrease in density and/or increase in pressure. So, 
$$
\rho_{\mathrm{air}}^{\mathrm{hot}} < \rho_{\mathrm{air}}^{\mathrm{cold}}
$$
and since the fire sword will locally heat up the airflow, 
$$ 
F_D^{\mathrm{fire\ sword}} < F_D^{\mathrm{normal\ sword}}
$$
A: Yes. Fire both experiences and creates drag. 
Your example of the fire sword confuses the issue a little and I'm not sure if it would be easier or harder because I'm not sure what a flaming sword is really. If its dipped in oil and lit on fire, it would be losing mass getting lighter, heating up the air around it and reducing pressure. But that is closer to a steady state answer; It might be harder to swing because the gas cloud around the sword would create pressure (like a rocket engine). This could be balanced because the cloud exists on both sides of the sword or it could be unbalanced because the moving sword would experience preferential combustion on the side getting fresher oxygen from movement. 
Let's use a different example to make things clearer:

Would a sword swinging through a flame experience drag?
Yes it would. A flame is a cloud of burning gas. As a cloud of gas, it creates drag. It also experiences drag itself. You can see this in that wildfires spread in the direction of prevailing wind, the flicker in a candle, or the motion of a mushroom cloud in a fireball. The reason the flaming sword question is confusing is that gasses react to drag differently than solids do. 
