Our steering column expert says to remove the lock tumbler.  If it stops buzzing, the problem is that tab on the tumbler.  Just replace the tumbler assembly.  If it still buzzes, it is the buzzer switch.  No one is yet reproducing them.  Any GM with T&T from 68 to 76 will work, but they are almost impossible to find.

Larry

But...you can't remove the key cylinder without removing the steering wheel. (except the '68, with the ignition in the dash)
Unless he knows some secret way to do it...and if he does, he should share...

There's a possibility that the wire for the buzzer has come loose from the door pin-switch, and is shorting against the body. That's the first thing that was mentioned, and the easiest to check. Just because the courtesy lights work fine doesn't mean the switch is not at fault, as there are two wires on that pin-switch...one for the lights, and one for the buzzer.
Adams' Apple2010-11-27 11:27:44

The black wire on the door switch is for the buzzer the 2 white ones are for the lights. I agree this is the best place to start.
The ignition lock cylinder has a tab on the bottom that closes the contacts on the key buzzer switch. When the key is inserted it pushes on the tab, I have seen these tabs get stuck from dirt and debris and keep the contacts in the closed position. This will cause the buzzer to sound only if the door is opened.
Posting the exact conditions of when the buzzer is operational IE: with door open/no key, with key/all the time, all the time/regardless of key or door, etc will help us point you to the best place to start looking.