Ok. The problem usually has to do with a MagicBand or old ticket being linked to a managed-by-you profile which can't just be removed from the system. I had to help my friend with this issue in trying to remove his ex-wife from his account.
You have two options, depending on whether you have (or want) contact with the person you're removing, though both are essentially the same: getting the other person to exist as a standalone MDE account so you can then defriend them.
The easier option if you're still in contact with them is to have them create an MDE account, you go into the listing on your account and fill in their email address to link to them. Once they accept, all Bands, tickets, etc move over to their account. You're then free to unfriend them as long as you don't have any future reservations together. This option would at least allow them to use their existing MagicBands again.
The second option is to create a dummy MDE account in their name, then become friends with them and unfriend them as described in option 1.
To do this, use a service like 10minutemail.com to create an auto-expiring email address. Use that temp email address to sign up for a new MDE account in their name. (You'll probably have to verify the email, so make note of how that temp email inbox works. You may have to extend it 10 or 20 minutes)
After that's created, go into your own account, fill in the temp email under their name. Go back into temp account to accept friend request. Then finally go back into your own account and unfriend the person.
After that you're done. The email address will go inactive, any emails to it from Disney will bounce, and the account will just sit inactive by itself with no friends. I'm not sure if Disney ever cleans out inactive accounts, but that's up to them. And if the ex-friend ever wanted to use their Bands again, they could probably press the matter themselves with the MDE Help Desk and get the inactive dummy account linked to them at a new email. (Probably easier to do if you set the dummy account up in their real name)
-Rob