Hello,
First things first, always make sure you have a battery plugged in to power the motors (since their power supply is different from the VCC on the rest of the board). That may help resolve any issues with the Motor TinyShield.
As for the RobotZero, the first released revision does not directly support DC motors. The next revision that supports DC motors has been designed, but not yet implemented in production.
You can, however, cut some traces on the board in order to allow DC Motor control. I have attached an image that shows the two traces you would need to cut on the board. You can make these cuts with a sharp hobby/X-ACTO knife. After these cuts, it would be difficult to solder them back together even with the proper equipment - so you would not be able to use stepper motors after this modification. This is a good solution if you only care about DC motors.
For using the DC motors after the modification, I attached an image showing which pins to use on the RobotZero (the two pins closest to the battery port with power on the left, and ground on the right in respect to the attached image) and an example program with a new DC motor function to control DC motors.
I hope that helps!
Thank you,
Réna