The oil level switch is probably a switched earth: ie if enough oil then the contacts remain open and no light, if not enough oil the contacts close and allow the bulb circuit to short to earth and the bulb lights up. If the light stays on then sounds like the circuit is permanent shorted to earth. problem is locating it. could be wiring, instrument panel, alternator even. Could be the cause of your flat battery

You gonna need a volt/ampmeter to check the current (amps) drain on the battery (ignition off) and start pulling fuses until the current draw stops (an 'ok' vehicle will generally draw around 30 milliamps [0.030amps]) To drain a battery overnight you will be looking at 1 or 2 amps drain. Once you identify which fuse circuit is draining the system you then have to start unplugging components on that circuit to narrow down further.