How to delete an old Apple ID on iPhone For a short time, I'm reverting back to my old iPhone.
I entered my current Apple ID (the one with email address). A long time ago, Apple ID wasn't an email address.

The pop-up "Sign in to iTunes Store" keeps showing. How can one delete the old Apple ID (not an email address) without resetting?
I’m not sure what you mean by resetting but if you go to https://appleid.apple.com/ from a computer you can manage Apple ID’s. 

Also here’s a link to an Apple discussion group with similar issue. 
https://discussions.apple.com/thread/5362309?language=en

Hope this helped.