If the voltage is zero, electrons don't move.
I'm guessing that you're talking about half-cell voltages here, that is, a single electrode in a single solution. You can't do anything with a half-cell. You have to join it with another half-cell to measure anything.
The most common example is the Danielle cell. One half cell is copper metal in a 1 molar copper sulfate solution. The positive half cell potential means that copper ions would 'rather' be reduced to the metal. The other half is zinc metal in 1 molar zinc sulfate solution. Zinc has a more negative half cell potential, meaning it would rather be oxidized to the zinc ion.
When the two cells are connected, one of them has to be connected backwards because the solutions have to be in contact (see salt bridge). For this reason, the two voltages are subtracted algebraically. IE the negative zinc potential -1.10 V is inverted to +1.10 V and added to the copper half cell voltage, +0.34, to give +1.44 V. The two metal electrodes can be connected to a voltmeter to read this voltage directly.
The voltage you might have been referring to is the voltage needed to prevent current flow from the cell. If you were to connect an outside voltage source to a Danielle cell, positive to positive and negative to negative, and increased the voltage, it would oppose the voltage from the cell until no current was flowing. At that point, the opposing voltage would be 1.44 V. Also, the reduction of copper and the oxidation of zinc would stop. If you increased the voltage further, the cell reactions would both run backwards.