I only had 2 semesters of physics, but I would think that an amp overdriven
far beyond its capabilites would create some distortion that would cause the
speaker to freak out and heat up.


Cause the speaker to "freak out"? Being a bit vague perhaps? As I
previously mentioned, heat in this case can only come from electrical
power. Not special waveforms. This is basically the underlying theme
of the first law of thermodynamics (kinda sorta).

No, the waveform has very little to do with anything as long as the
frequency is high enough to allow for the cone motion to sufficiently
move air and dissipate heat (anything over a few Hz is fine for this).
Power content has everything to do with it. The speaker doesn't care
whether the increase in power is due to clipping or due to just using a
more powerful amp. The only thing the speaker cares about is how much
current is flowing through the coil. That's it, nothing more.