You can stay signed in to OneNote indefinitely, but if you want to change accounts or are using a shared iPad or iPhone, you can sign out of OneNote.
In OneNote, tap the Settings icon next to the account name.
On the Settings screen, under Account, tap the account name you'd like to sign out of.
Tap the account name you'd like to sign out of and then choose Remove Account.
Accept the notification that appears to confirm signing out.
Note: You can close any notebooks you're not using (or not signed into) if you don't want to see them in the list. Swipe left on the notebook you'd like to close, and then tap Close.