Could you instruct a car thru the OBD port to open the trunk even if the car was turned off?

According to this response, when the ignition key is removed from the car it's usually not possible to communicate with the car via the OBD port so assuming by "turned off" you mean the ignition key removed, I'd say in most cases it is not possible.

The whole idea of the On Board Diagnostics port is to provide vehicle diagnostic information. It is read-only, except for specific log entries and specific fault codes that can be reset. It is not meant as a control interface for your car.

Also see Can i control the brakes of a car by the OBD II Interface?

I've seen someone with a diagnostic tool (sort of like the one your mechanic uses, that has its own power supply) tell a car to open its trunk via the ODBII port. The customer had locked their keys in the trunk and a roadside tech could open the door, but the security disabled the trunk release.

Being that roadside tech, I would dearly love to find such a tool. Currently I have to open the car manually, then disassemble the door (on some cars) to access the lock and 'unlock' the car by manipulating the lock manually. A PITA, to say the least.