Would OBD-II devices drain the battery when used 24/7?

If the engine isn't running, then they surely will. I have an ELM 327 (Bluetooth) device which plugs into the OBDII port. Even when the vehicle is turned off, the device is active. Before I realized this, I had left it plugged into my car, came out the next day and the battery was dead. Inside the OBDII port, the power lead there is always on and gives voltage straight from the battery. If there's power there, the OBDII device you have hooked to it will draw power, so will drain the battery.