When I right-click on the Office 365 Word icon in the Windows taskbar, I get a list of recently used Word files. When I right-click one of those filenames and select “Remove from this list,” that filename disappears. But if I shut down and restart the computer, it reappears. How can I delete certain filenames permanently from that list?
(I tried deleting them from C:\Users\[my name]\AppData\Roaming\Microsoft\Windows\Recent Items but that didn't work.)

Afaik Office will always display the number of Recent Files you've chosen to show in the options. I suspect the only ways to make some filenames disappear are:
1. Reduce the Recent Files number in Options;
2. After deleting as you did, open enough other files so that they become more recent—you probably need to make a change to such files, so they'll register as Recent, my guess is the file modified date drives the show.

If you're comfy in the Registry, you can look in the various hives for eg:
Software\Microsoft\Office\16.0\Word
There are a couple of MRU keys usually, but they don't look friendly to me so I've never messed with them.