Just wanted to follow-up with the outcome of my situation, in case anyone else might ever run into a similar unidentified "chirping" noise in the future.  Almost immediately after my original post, the Dell Support folks reached out to me to try and help.  We worked together over the course of several days trying to troubleshoot various possibilities for the source of the noise (things like fans, video card, drives, power supply, etc.).  Even after trying all of the things we could think of, my ears were still not able to isolate exactly where the noise was coming from, so at that point, the Support folks recommended that we set up a Mail-In service to have one of their techs check it out. (Side note:  Aside from a minor delay due to supply-chain issues, the entire mail-in process was surprisingly pleasant.  They sent me the necessary packing materials, shipping in both directions was very quick, and the communications were great throughout the process.  I was very pleasantly surprised with the whole thing.) So anyway, the service was done and ultimately they ended up finding that the source of my "chirping" noise was something to do with the Heatsink/Fan.  Those parts were replaced under warranty, and the PC was shipped back to me.  It's been back at home, running in my quiet room for several days, and has not made any more unexpected noises. Thanks to everyone involved, for all of the assistance.
I've seen a few "chirp" related posts, but nothing exactly like my issue.  I bought a T5820 from the Dell Outlet several months ago.  A few days after I received it, it started making a quick 'chirping' noise every 10-20 seconds, kind of reminiscent of a smoke detector when the battery is dying.  All updates have been loaded and all diagnostics come back clean.  The noise happens whenever the machine is powered on.  It happens when fully booted into Windows, or when just sitting at the BIOS/UEFI screen (it starts chirping even before it gets that far). I added a couple of new SSDs to the machine when it first arrived, so I ejected those to make sure they were not the cause.  That did not make any difference.  I also ejected the boot SSD that came with it, and the chirping noise still happens even when there are no drives attached at all. Since it reminded me of a smoke-detector, I replaced the CMOS battery.  That did not fix it.  I thought it might be the power supply, so I bought a couple refurb PSUs from eBay and swapped them in just as a test.  Neither of those made any difference either.  Still chirping. The machine came with a NIC and video card, so I tried removing those, but the chirp still happens.  So from the best I can tell, it's not related to anything with software, drives, or peripherals. I don't know how to boot up the machine with the side cover removed, so I have a difficult time pin-pointing exactly where the noise is coming from.  The best I can tell is that it originates from somewhere on the motherboard, near the area of the back USB ports.  I can't think of anything else that I can check.  The machine still functions normally, but trying to use it in a quiet room is kind of like that water-torture, but instead of "drip-drip-drip" every few seconds it's "chirp-chirp-chirp".  I hate to admit it, but I've gone back to my old PC for now. Has anyone else had a similar experience?  Any ideas how I should proceed?
5820 chirping noise