The most likely suggestion is that the fuel pump relay contacts are stuck. Switch the ignition on (don't try to start), wait 30 secs, remove the fuel pump relay.
The pump should stop.
If the pump stops, insert the relay again and listen for a "Click".
No "Click" but the pump runs = faulty relay contacts.
"Click" & pump runs = possible faulty PCM (Powertrain Control Module) or wiring from PCM shorted to ground.
FYI, the low oil pressure scenario should cut power to the pump and not allow it to run continuously.
HTH.