Does the lift pump in the tank run continuously while the engine is running
cant speak for every car out there, but 90%~ of the time, it is on if the following conditions are all met.
A: the key is in the run position and power is switched at the ignition provided fuses and relays are operational, 109 relay, etc...
B: All safety mechanism is in its open position. Airbags are not deployed, car is not upside down, etc......
C:The battery has sufficient power to run the pump.
Some newer cars like hybrids are some of the exceptions to this as they turn off the engine while driving or at a light depending on power requirements.