I had a broken turn signal switch. I bought this replacement. Now 2 years later I get my car on the road and I notice that when the door is open, the door buzzer, buzzes. Removing the key used to silence the buzzer. So to keep it short, the issue was the turn switch. Well kinda.
Part of the turn signal switch is a wire that connects to the ignition switch (to detect the key). That wire is held in place by a white plastic piece that fits inside a little pocket in the steering column.
Look at the contact that is supposed to touch the ignition switch. If it gets bent via shipping or mishandling, it can touch the steering column pocket instead of the ignition switch. Since the steering column is grounded it acts like the key is present even when it isn't. Simply bend the contact back to the center of the pocket and you should be good to go.