Is it a good idea to ground a high voltage DC motor?
Is this a good idea, a bad one or an unnecessary one?


It's likely that there will be some path through the case of the motor to the lathe to the controller anyway — so adding the ground wire will just make that more guaranteed and predictable, which is good.

Two general good reasons to ground a motor:


General electrical safety as with any device having an exposed metal chassis — redirecting any short-to-chassis to definitely go to earth rather than someone touching it. For this purpose, the ground wire (and other ground connections all the way back to the outlet) should be able to safely carry short-circuit current from the motor controller (for the short period until the circuit breaker opens) — it may be advisable to connect it specifically to an intended ground terminal on or in the speed controller rather than just to some point on the housing.
Shielding against RFI generated by the motor. (Another common practice, at least for small DC motors, is to add capacitors between the case and each motor terminal. I can't say whether this is a good idea in this high-power application or what capacitor specifications you would want.)



  Also, I usually test my projects using a GCFI and I wonder if there is any possibility of inducing a current in the ground wire which can trip it. It shouldn't, as the ground is an incomplete circuit, but I thought I should ask anyway.


The concern here would not be inductive but rather capacitive coupling, between one line and ground. This could trip a GFCI but if there was enough to do that then it would be enough that you would want to do something to reduce it anyway.

(And yes, I did just mention adding capacitors above — smaller ones.)