Carry a helium or hydrogen filled balloon in a car, and accelerate the car. (Be careful; this can be dangerous.) The balloon will not be pulled to the rear of the car; it will move to the front! (That's why I said you have to be careful.)
The balloon is, of course, pushed backwards by the acceleration (according to the equivalence principle, this is equivalent to rearward gravity), but the air, since it weighs more per volume, is pushed more. So it is what goes backwards, while the balloon is forced forward by the heavier flowing air.
Likewise from gravity. A balloon goes up because the heavier air falls down around it, displacing the balloon. And similarly for the hot plasma known as fire. It weighs less per cubic centimeter than does the air around it, so the air drops and pushes the plasma up.
When you start a fire, fuel combines with oxygen to release heat, light, carbon dioxide, water vapor, soot etc. The heat energy generated in this process heats up the air around the flame which in turn decrease the density of the surrounding air.
This hot air around the flame then starts rising up and the surrounding colder heavier air rushes down into its place accelerating the hot air upwards which in turn causes the flame to shoot up. It is this hot air rushing upwards that also causes fire to flicker.
In case of in a micro gravity environment, there is no reason for the heavier air to fall down and rush into the place of the lighter air. So the flame just remains a spherical blob as you would imagine.
Thus, gravitational pull exist on fire too.