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That has to be a faulty DRL module. The DRL module (day time running light module) is supposed to turn the lights on/off as you turn the car on/off. If the lights are staying on, that means that the DRL module is not turning them off. Also the noise not being there is not relevent, because the noise should not be there when the DRL stay on, because they are not supposed to stay on.
Do the other sounds work?
(such as when you drive and take the seat belt off, does it make the beeping noise?)
Okay here is the deal.
The beeper is fed by a signal from the tailight relay to the body control module that in turn signals the gauge module that sounds the beeper.
I would have to say if everything else looks ok-door switches ok, key beeps when you put it in the ignition, the tail lights work, then the fault is in the module .