How to remove old Apple ID

Have you checked to make sure you are signed out of everywhere on the phone? Are the popups for update from the App Store?

Check the App Store and iTunes Store to see if there are any updates for apps/purchases. A request for another Apple ID password is often caused by updates to apps/purchases that were made when using the other ID. Those purchases are tied to the other ID and can't be transferred. One option is to delete the apps/purchases made by the other ID and download them using the new ID if desired. After deleting the apps, it may be necessary to restart the device/computer before you can download the apps again. If necessary, check your purchases history to see which apps/purchases weren’t made by the current ID. 

Purchase History Review.  *

Or you can try to set up Family Sharing  between the IDs to keep access to your purchases.

Family Sharing.         

Family Sharing - Set up.        

You are welcome. Glad to help.