Can undervoltage kill a circuit?
Yes.  Some circuits are non-linear, or particularly, anti-linear: when voltage goes down, current goes up.  Notably, boost converters which pump up supply voltage to get their working voltage (e.g. some wireless mice use a single AA battery and run fine on 1.2v NIMH's, or the apocryphal Joule Thief.) As voltage falls, they must increase the current they draw to produce the same total power (watts = volts x amps).  Depending on how the circuit is specced and protected, you could exceed the current limit of one of the components.  

High voltage just requires thicker insulation, which is quite easy, especially in the low-voltage range of Arduino devices. High current requires bigger pipes: wider traces (which gobble up precious circuit board space), thicker wires, more silicon on the chip, thanks to Ohm's Law more heat dissipation, etc.  Which is harder and costs real money.  As such, devices tend to be more sensitive to overcurrent than to overvoltage, especially when the cost-engineering is done.