Must an LCD panel meet specific requirements to support a higher refresh rate?
The driver circuitry for the panel is responsible for pushing frames to the panel at a given rate, but are there any particulars of the LCD panel itself that factor into achieving the higher refresh rate?

Since the panels are big 2D arrays of pixels addressed similarly to a DRAM cell, the LCD itself must be able to clock at the pixel rate you're driving.  If its internal capacitances are too large or its transistors too slow, there won't be time for the pixel to update before the next value is driven into the array.
In addition, since the pixel array is essentially an optical digital to analog converter, there are also sampling rate concerns.  Driving the array at a higher pixel rate is equivalent to increasing the temporal sampling rate driving each pixel.  Increasing the update rate while keeping the analog impulse response of each pixel unchanged is equivalent to increasing the sampling rate of a DAC without also adjusting the analog bandwidth of the electronics it is driving.  Eventually your analog bandwidth falls far enough below the Nyquist (folding) frequency that although you are updating the pixels more frequently, no new information is actually being modulated onto the pixel output because it simply cannot respond faster (even if the array itself can physically sustain the pixel rate).
So yes, there are two factors.  The array must be able to shift values at the pixel rate you've selected and the pixels themselves must be fast enough (have sufficient bandwidth) that the faster refresh rate actually does something.

In other words, is it theoretically possible to drive a panel over its specified refresh rate using another driver board, assuming all interfacing requirements are identical?

Yes, and you do occasionally see manufacturers offering displays at refresh rates above the nominal spec value for the panel they are using.