Went for the unorthodox solution: a small piece of electrical tape applied to the top of the beeper.  It blocks the hole in the top of the resonator case of the piezo buzzer, itself, and almost completely stops the beeping noise.  So the beeper is still beeping away merrily, I'm sure, but I can't hear it anymore.  It's unfortunate that I was not able to get to the root cause of the issue, but such is life.
Hey-- I recently got a used Optiplex 7020 SFF to mess around with.  Yesterday, I noticed a beeping noise  in the room where I have it.  The sound was so faint I thought it was coming from outside the house.  Noticed it again, today, and took about an hour to track it down to the power supply of the machine.  It's so faint, in fact, that I had to put my ear right next to the open case to figure out where in the machine it's coming from, and I found that it seems to be coming from the PSU.  But the oddity has just begin.... Anyone have any ideas about what that could be?  Honestly, I'm not all that worried about it, as I already have another power supply on order, a 315 watt unit from an Optiplex Xe2, as I'm putting an SSD boot drive, a big secondary hard drive, a small graphics card and a Core i5-4590 in this machine (it's destined to be a home entertainment machine, hooked up to my TV).  Thing is, the only other time I've heard a faint beeping noise coming from a PSU, it was the outboard power supply for a Toshiba laptop, which proceeded in short order to burst into flames in my hand.  Ah, memories....
Optiplex 7020 SFF PSU Beeping Faintly