Neither. I am not familiar with your car, but the fuel pump should typically only run for one or two seconds after the ignition switch is turned on, then again when the engine is rotating, meaning cranking or running. If the engine stalls or is stopped by you, the Engine Computer will see the lack of signal pulses coming from the crankshaft position sensor and / or camshaft position sensor, and will turn the fuel pump off in response.
If the pump is running, it is okay. Leaking injectors are always a possibility, but that wont make the fuel pump turn on. The pump does not react to fuel pressure.
The place to start is to look for a fuel pump relay. Pull that out and the pump will stop running. If it does not, we have a very unusual problem. Swap in a different relay to see if the pump starts up again. If it doesn't, the old relay may have contacts that welded themselves together. A different relay will be the solution. If you feel the replacement relay click on and the pump does start running again, the first suspect would be the Engine Computer. That is really not common either so I would prefer to look at a wiring diagram to verify that before buying a replacement.
Also consider that on some car models, the fuel pump is turned on by a tap on the oil pressure sending unit. If you have that circuit, a defective sender that is stuck on, or a grounded wire going to it could keep the fuel pump running.
The purpose of all of these circuits is to stop the fuel pump if a fuel line is ruptured in a crash. Regardless of how it's done, an engine can't run without fuel pressure. A stalled engine will not generate position sensor signals, and / or there will be no oil pressure. Either way the fuel pump will be turned off so no raw fuel gets pumped onto the ground where it would be a fire hazard.
Wednesday, June 8th, 2016 AT 8:58 PM