What is the voltage range of a standard headphone jack from a phone?

Commercial line out specification is to be able to drive 1 milliwatt to a 600 ohm load. For a sine wave, this means a voltage of 0.77 volts RMS (2.2 volts peak-to-peak) and a current of 1.3 milliamperes RMS (3.6 milliamperes peak-to-peak).

Check out: http://en.wikipedia.org/wiki/Line_level

The most common nominal level for consumer audio equipment is −10 dBV, ... Expressed in absolute terms, a signal at −10 dBV is equivalent to a sine wave signal with a peak amplitude of approximately 0.447 volts, or any general signal at 0.316 volts root mean square (VRMS). ... There is no absolute maximum, and it depends on the circuit design.

This is however for the "Line out" plug which, apparently, carries a signal at a fixed amplitude and lets the receiving end determine the volume.

In most cases changing the volume setting on the source equipment does not vary the strength of the line out signal.

For a speaker-driving headphone plug I believe things might get more complicated, since that signal is really rather a current signal (used to drive the coil of a speaker).

In contrast to line level, there are ... those used to drive headphones and loudspeakers. The strength of the various signals does not necessarily correlate with the output voltage of a device; it also depends on the source's output impedance which determines the amount of current available to drive different loads.

I guess your best bet might be to look at the wave with an oscilloscope, which should have a high-impedance input like the Arduino's analog input (ADC).

(I'm no expert, take with a grain of salt and feel free to edit)

Edit: The Wikipedia article I used as a source has been edited a lot since I originally posted this answer. Among other changes, the qouted pieces above have been removed/changed. Therefore I'm striking most of this answer out and recommend referring to the Wikipedia article linked at the top.

Unfortunately there is a lot of "audiophile" nonsense around headphone amplifiers and headphone impedance. Probably the top 5 results for "headphone impedance" on Google are just wrong. This site contains some useful information (though a lot of it is wrong too).

But anyway if you look at the graphs which I assume are correct, you can see that in the audio frequency range most headphones have a fairly small reactance compared to their resistance. And most headphones have an impedance around 16-32 Ohms with some crazy "audiophile" headphones having higher impedance (e.g. 300 Ohms). He suggests that 5 mW is sufficiently loud for portable headphones. Audiophile headphones will require higher power.

Power is \$P=V^2 / R\$ so \$V = \sqrt{R*P}\$, so high impedance headphones will need a much higher output voltage because they require more power and have a higher impedance. Anyway, for the Sony MDR-EX51 headphones shown on the page linked above you can see that they are fairly close to a simple 17 Ohm resistor. At 5 mW that would mean a voltage of 0.3 V and a current of 16 mA.

An Arduino can supply this fairly easily but I don't think you can just hook it up to PWM since 5V across 17 Ohms gives 300 mA which is well above Arduino's 25 mA limit. A simple solution may be to insert a 4.7 V / 16 mA = 290 Ohm resistor in series with the pin.

I haven't tried any of this - you'll have to experiment!