A balloon with mass M is descending down with an acceleration a where a < g. What mass m of its contents must be removed so that it starts moving up with acceleration a?

The balloon descends with acceleration a < g.
Thus, the force experienced by the balloon is in the upward direction and is less than the weight Mg, which is acting downwards. 

                
Therefore the equation of motion of the balloon is,
                   Mg - F = Ma                        ...(1) 
If some of the mass of the balloon is removed, the downward force i.e. weight of the balloon decreases.
As more and more mass is removed, the downward force on the balloon decreases more and more, and becomes equal to upward force F.
On further removal of mass from balloon, the upward force F exceeds the weight of balloon and balloon starts accelerating in upward direction.
Let, mass 'm' be removed from the balloon. 
Upward acceleration acquired by balloon = a 
Now the equation of motion of balloon is, 
F - (M - m)g = (M - m)a                       ...(2) 
Adding (1) and (2), we get
              mg = (2M - m)a
            
Mass 'm' given in the above expression should be removed so that the balloon starts accelerating upwards.