Motherboard backside traces touching antistatic mat. What happens?
What happens when an unpowered board (and completely disconnected from
  any peripheral) is on a grounded antistatic mat?


All potentials on the board settle to zero (ok maybe not the battery and RTC circuit) but they would float to whatever ground is (which is close to 0V)
It would be almost like attaching a 1MΩ resistor to every point on the board and then to ground. If there was air in between the board and the mat it would be even higher.







simulate this circuit &ndash; Schematic created using CircuitLab


  Will the ground-plane reach equipotential with the mat, because the
  GND traces make contact with the mat?


Eventually it will, depends on how much charge exists on the motherboard. which should be really low. 

The mat is 1MΩ (or more, I just measured mine and across the whole mat is 10MΩ to the button, yes I realize its not super accurate, but its close) to ground, most mats are made of dissipative material so this also won't affect the high speed signaling (noticeably anyway) if you run the motherboard right on the mat. 

We do have ESD tables and they are 1MΩ also but underneath that thin 1MΩ layer there is a conductive metal layer. I would not run a motherboard on this because of a risk of capactivly coupling the signals to ground or to each other. This probably wouldn't affect the operation of the board but it could so I would not do it.