FYI see the update to this post here. The gas furnace in my house is only 7 years old. A couple weeks ago, it started singing to me. Normally a baritone, the furnace decided to expand into the tenor vocal range. Then a certain pain was evident in its song until I finally translated it as “help me . . . help me . . . help me.”
Sure, I could call the number on the side of the furnace and get charged for premium weekend service, but I decided to dive right in and see if I could fix this myself. After all, how hard could it be? I have the internet plus an awesome set of tools.
I started by simply removing the access panel and observing and listening. From there, I found several different manuals available for download via the web and pinned the noise down to the draft inducer motor. Apparently somewhat common, the sealed bearings were worn, causing the shaft to bind and cause the noisy squealing.
I called my friend Steve, who dropped by and we pulled the motor and put a temporary fix in place. We unsealed the bearings, cleaned them, and re-packed them with lithium grease and reassembled. Lo and behold, no more noise! I found the part online for less than $100 and will replace it when it arrives.
I can imagine a service call would have resulted in one charge, then the part would have been ordered and the next service call to install the new motor likely would have totaled over $500. 
The draft inducer motor is just behind the plastic cooling fan above.

The part that is to be replaced is the motor (the black part) Carrier part number 318984-753.
I found it online at Shortys HVAC Supplies.