No, the fuel pump on a 1990 Pontiac Lemans (and most fuel-injected cars) is
not supposed to run continuously when the key is simply turned to the "on" position. It should only run for a few seconds during the initial key-on, to prime the system, and then shut off. It will then run only when the engine is cranking or running.
If the fuel pump runs constantly with the key on, there's a problem in the fuel pump circuit, possibly a faulty relay, a short in the wiring, or a malfunctioning computer module (PCM). This is a serious issue and should be addressed to prevent damage to the fuel pump and potential fire hazard.
