For months, I had some phantom folders stuck in Notes. Every time I deleted them on my iPhone, iPad, or even iCloud.com, they'd come back a few minutes later. I tried every combination of possibilities to try and remedy this, from turning off syncing completely to deleting folders on every device simultaneously—but they just kept coming back. The only remaining option was to erase all Notes data from iCloud.
I could have left the folders alone since they were empty, but I keep Notes organized and their existence was an annoyance. Before I wiped all my notes from iCloud, I wanted to back them up. I enabled the "On My Mac" account in Notes on my iPad, recreated the folder structure I had in iCloud, and moved all my notes1. I then deleted all Notes data from iCloud from the iCloud Storage options.
After confirming all Notes data was erased, I recreated the folder structure back on iCloud, and moved my notes back. The phantom folders had been successfully exorcised.
I'm still not sure what caused the problem, and likely never will, but it started not long after I began using the public beta of iOS 11. Regardless of what caused it, deleting Notes data resolve the issue for me.
I usually delete notes I no longer need, so I only had about 60 notes across eight or nine folders to move in total. ↩