One of the two black wires going to the seat belt runs to the buzzer, the other to a ground under the kick panel emergency brake side. The buzzer is interesting, I someday would like to figure out how it works inside, but basically if the black wire completes a ground (when the seat belt is clicked into the buckle), then the buzzer is grounded and will not buzz. It's like a mini relay with a timer. If it's not grounded, the buzzer will buzz for x seconds and the seat belt light will light up.
I suspect your buzzer timer is bad, because I think the seat belt light is supposed to turn off when the buzzer stops buzzing. I may be mistaken though.