On an iphone how do i log out of an apple id that no longer exists

If it no longer exists, meaning you followed these directions→How to delete your Apple ID account - Apple Support. If this is what you actually did and did not log out first that iPhone is now useless, as you were warned when you deleted the Apple ID.

If you mean that the email address is no longer in use, you can still use it as the user ID for your Apple ID account, enter it and enter its password to disable Activation Lock.

Use the Get Support link at the top of this page. They may be able to temporarily restore the account.