Could you instruct a car thru the OBD port to open the trunk even if the car was turned off? Can you turn the accessories on from the port then open the trunk?
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.