I'm attempting to conduct an experiment with a balloon. I fill it up with some amount of helium. If I cool the balloon down (say with liquid nitrogen), the balloon's volume will decrease (because of Charles' Law), and if I heat it up, the balloon will expand. Does this expansion and contraction of the balloon mean that it will float more or less? Or does the amount it float just correspond to the mass of the gas contained in the balloon (in this case constant)?
The force on an object immersed in a fluid due to buoyancy is given by
$$F_B = \rho V g$$
where $\rho$ is the density of the fluid, $V$ is the volume displaced by the object (the volume of the object itself, if it's fully immersed), and $g$ is the acceleration due to gravity.
You can see that this doesn't actually depend on the mass of the object itself at all, only the mass of the air that it displaces. Although of course the object's mass does come into play when you compare the buoyant force with the downward force due to gravity, which is proportional to the object's mass.
So in your scenario, you will decrease and increase $V$ as you cool and heat the balloon, and thus decrease and increase the buoyant force, changing the acceleration of the balloon. You could even create a scenario where the balloon falls downward when cooled but ascends when heated.