I had the same issue on my work computer and on my android phone. I use my personal Microsoft account for OneNote across all devices. On desktop devices, I use 'OneNote for Windows 10,' rather than OneNote 2016. I prefer the user interface of OneNote for Windows 10.
OneNote had a message popping up saying it was working in offline mode and changes would not be sync'd unless I signed in. But clicking the "Sign In" button did not result in any pop-up window. Above that 'Sign In' button, my account name was there, showing that I was signed in, but the message did not disappear.
I signed out of my account and closed OneNote for Windows 10, then re-opened it and signed back in. My notebooks were not showing on the left side of the window, so I clicked "More Notebooks" and chose the notebook I was trying to use, then clicked "open" in the "Choose Notebooks to open" dialog box.
This added the notebook and all its sections, but none of the pages. And now, when I click on a section in my notebook, I get the message "You don't have permission do access this section. We'll sync again if permissions are restored."

I do not share my notebooks between any accounts, and I do not use any shared notebooks. This notebook is mine and mine alone, so I do not understand what would cause a permissions issue like this.
Note that the "to sync this notebook, sign in to OneNote" issue has reappeared. The Sign In button still does not do anything.
I then attempted to check OneNote on my Android phone (Galaxy S10+), only to encounter the exact same sequence of events: [Working offline, please sign in] --> [Already signed in] --> [Sign out] --> [Close, reopen, sign in] --> [You don't have permission to access this section].
I am able to use the "OneNote" app on my work PC without these errors, but not "OneNote for Windows 10." The "OneNote" app looks like this:

I do not have this app on my home PC, and it does not exist on android. I need to be able to access my notes, but all available microsoft support phone numbers have removed all of their tech support personnel and you cannot reach a real person - only a recording instructing you to look on the forums for answers.
Igor, I hope you can help with this issue. I have not seen this problem in my 8 years of using OneNote for Windows 10. Please let me know if there is additional information I can provide.