This is new to me but I guess it is possible. Looked at the wiring.
The power for most if not all engine components is supplied through the main relay. This includes the O2 heater. If the main relay is engaged, the VST pump will run. The ECU sends a ground to the relay to engage it. So my first advice is to see if the VST pump has power when the ignition is turned off. Same for remote oil pump.
If the main relay is engaged all the time, I would expect your battery to run down fairly quickly when sitting.
The signal wire to the main relay is Blue/yellow and goes from the ECU main large connector directly to the relay (through a harness of course). If, when ignition is OFF, that lead measures 0 volts relative to a good ground ( in other words it is grounded) either the ECU is bad or a wire is shorted. If it reads 12 volts when the ignition is off, the problem is not there.
