I'm using a Note 8 with a rather thin case with the OM2. It doesn't seem to heavy for the OM2 but can feel a tiny added weight with the case. My question is do many of you leave cases on your phones when mounting on the OM2? My long term concern is with the motors.

DJI states you should not leave it on.
Just because the need to be on the safe side when it comes to warranty claims.
I know some cases don't weigh too much, but others will and it is hard for DJI to make a statement for what will work and what won't.
So if you want to be 100% sure, get it out of the case.
BUT
If it balances properly and the motors don't get warm you will probably be fine.
And I can see the inconvenience of removing it all the time.
But don't sue DJI when it breaks

Montfrooij explained very well that we do not recommend to mount the phone case since for larger phone, the case will affect the balance.