Will the Stepper Motor 17HS3401 work fine with the TMC2130 driver? Or is the driver chip going to burn? Are they compatible?
Higher operating voltages translate to higher stepping rates (and higher peak RPM, higher torque at higher RPMs).  If you operate the motor from 3.12V, you will get the rated torque at 0 RPM (aka holding force), but performance at any speed will be poor.  Since the TMC2130 driver is a constant current PWM driver, higher voltages (up to the rated voltage of the TMS2130 and any filter caps/etc) will produce better motor performance.  The TMS2130 chip may get hotter (due to the internal LDO to drive internal circuitry).

For most applications you will be fine with 12V.

Additional simplified explanation:  The higher voltages are needed to quickly change the current flow in the motor coils