Would a balloon with a heavier or lighter gas fall faster?

Newton's law of universal gravitation has since been superseded by Einstein's theory of general relativity, but it continues to be used as an excellent approximation of the effects of gravity. Relativity is required only when there is a need for extreme precision, or when dealing with gravitation for extremely massive and dense objects. Also buoyancy and fluid dynamics has a part to play in this scenario.

Your example is dealing with buoyancy vs gravity in liquids/gases, e.g. half filled helium balloon vs quarter filled helium balloon in air. Once the two balloons both have negative buoyancy, then fluid dynamics for gases/liquids comes into play, and the more negatively buoyant balloon will sink faster.

Heavier, because it would pull the balloon down.