Is it possible to damage a phone by charging it with an intermittent power source?
Probably not. Lithium batteries have no memory effect have no problem handling a bit of charging followed by a bit of discharging.
For example, by using piezoelectric transducers in the soles of your shoes.
Now, this is interesting. If you somehow manage to create a 5V supply with enough current, then the phone will start charging.
Since the power generated by these piezo elements will be ridiculously small, it is highly likely that as soon as the phone starts drawing current to charge itself, your device's output voltage will collapse, and the phone won't charge at all.
However, the screen will light up for a few seconds and display "charging..." and this will most likely use a lot more energy than what your piezo elements can provide.
Therefore, the net result will be that the phone's battery will discharge quicker (due to the backlight) than if it had not been plugged into the sneaker-charger ;)