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.

Typically when you change one (voltage or current), the other will change as well (except with infinite resistance ie. open circuit or zero resistance ie. superconductors).

But speaking for speakers, most of which consist of a magnet and a big wire coil connected to a paper cone, it is the current flowing through the coil which creates the force that moves the coil and cone and produces the sound you hear.

Prompted by a comment by @BruceAbbot (on a previous answer of mine that I deleted because it wasn't spot-on) I did some further researches and I found a reference that seems perfectly fit to answer your question.

In short: modern audio power amplifier generally have a very low output impedance (fraction of ohms) and act as (almost ideal) voltage sources. Therefore, using your terminology, they "modulate" the voltage across the speakers, which react absorbing the current needed for their operation, as their characteristics mandate.

The reference is Audio Power Amplifier Design Handbook, by Douglas Self (link to google books), under "Damping Factor" section. Excerpt:

Audio amplifiers, with a few very special exceptions, approximate to perfect voltage sources, i.e. they aspire to a zero output impedance across the audio band. The result is that amplifier output is unaffected by loading, so that the frequency-variable impedance of loudspeakers does not give an equally variable frequency response, and there is some control of speaker cone resonances.

The few exceptions cited are trasconductance power amplifier used for so-called current-driven loudspeakers, where the amplifier act like a current source and "modulates" the current into the loudspeaker, which reacts by generating a voltage across itself accordingly.

See also this EE.SE answer: How important is impedance matching in audio applications?