We have a way to purge data from views, a way to purge data from Core Data, a login screen view controller to present, and a way to delete our session object.
Now undoubtedly we'll need these together to logout and show the login screen. Should there be a function in the appdelegate, or another class specifically for this functionality, or handled individually by wherever we need to logout from?
What's the Objective-C convention for this?
You have a user object, or a user controller to manage this functionality, right? So that should do it.
The app delegate is for app level event handling, not user management.
Objective-C is a language so it has no convention for user management.
Duplicating the logout code "wherever needed" is bad design. This is really a pure OOP question and you need to define classes and allocate responsibility and ownership accordingly.