How do I unlink my Facebook account from the McDonalds mobile app?

Short answer: At the time of this article (Dec 2021) YOU CANNOT.

You can’t even change your e-mail address with the McDonalds app even if you signed up using your e-mail address only.

According to the official McDonalds Mobile Order app help page:

Can I change the Email Address? No.

I suspect that the developers for McD decided to use the e-mail/login-method as the primary unique key for each user instead of assigning a number for the userID. Kind of ridiculous. If you get fired from work or graduate from school, your e-mail address is going to change. If you move from Meta to Google, your login method is going to change. McDonalds expects each customer to possess a “for-life” e-mail address to be using the mobile app.

I hope this changes in the future because it is totally do-able technically and makes sense practically.