Wouldn't overdriving any lamp do damage to the ballast by overheating it?
Depend on ballast concept and what the overdriven lamp does.
MV will not do anything to the ballast, as it's voltage is stable and the same for nearly all wattages, so the ballast would "think" it has correct lamp in it. Of course, the lamp will explode soon.
SON (HPS - for Americans :-)) would most likely cycle - as overheated tube rises the sodium pressure, so discharge voltage, till it exceed (with restrike spikes) the ballast OCV (except low voltage run on high voltage ballast - e.g. US HPS70W run on HPS150W ballast). The ignitor will quickly eat-up it's lifetime (as it's designed for only few pulses per start).
MH: Unsaturated vapor concept (some QMH) behave same as MV, saturated vapor concept (all CMH, some QMH) behave like HPS.