Once the fluid from a pleural effusion is mostly removed, will the rest go away by itself?

It depends. It depends on the cause of the effusion and what the effusion is made up of. The effusion should be analyzes for both cells (cytology) and chemical analysis. You will then need to discuss the results with your doctor.
Yes. Your body has lymphatics in the space between the lung and chest wall. These function as a plumbing system for the body and help to reabsorb fluid from the pleural space (where the pleural effusion originates). The rate of absorption is slow but your body should reabsorb the fluid that is left behind in the space.