The absolute answer would depend on a lot of factors, but the basics of it would be that - Yes, the volume does change and Yes, it would have an affect on the membrane potential.
By adding mass to any liquid solution, you are changing the volume. Plain and simple. Liquids are not compressible, and the only way to maintain volume while adding mass would be to increase the density of the liquid. [Edit] I'm wondering if you don't mean increase the volume of the cell - in which case, in extreme conditions where the cell cannot maintain water and salt concentrations, the cell would lose volume as water and salts move out. In a hypotonic solution (much more water than solutes), the cell would lyse as osmotic pressure favors water moving into the cell
The membrane potential would change, but how much depends on a lot of factors. It would most likely change due to the movement of ionic solvents down their concentration gradients. Osmosis would move water out of the cell and into the extracellular environment, and ultimately reduce the membrane potential as more water outside the cell would essentially decrease the molarity of the solution. The cell would have to pump more ions out in order to regain the membrane potential.
The exception might be if you added so many hydrophobic molecules outside the cell that you basically saturate the environment, which could prevent exchange of all charged or partially charged atoms and molecules.