is sound produced by varying current or voltage?
There are two main ways electronics can generate sound:

As Dave said, some speakers use variable current to create a variable electromagnet which moves the cone and produces sound (of course, you need some voltage to induce the current).

Another way is with a piezoelectric material. When you apply a voltage to certain materials they can mechanically change shape. By varying the voltage, you can produce vibrations, and thus sound. Again, these devices must consume some current, but this is can be small compared to the voltage.

In summary: All electronic sound sources must have current and voltage to operate because they convert electrical energy into sound (and heat) energy. The electrical power they consume is given by \$P = I V\$, where P is the power, I is the current, and V is the voltage. In piezo devices V may be quite high and I quite small, and similarly in electromagnetic devices I may be quite high and V may be relatively small. However, if either of them is 0 you get no power, and thus there can be no sound.