I can’t get rid of an old iTunes account password popup.

Have you checked your purchases history against all the apps/music on the device to make sure there isn't an app/music from the old ID that is still installed.

 Purchase History Review

Have you tried signing out of the App Store/iTunes and then back in?

Try a restart.

Restart your iPhone, iPad, or iPod touch

Try a forced restart.

Reset Device

I have a similar problem. A few apps (I don't know which ones) on my iPad were originally downloaded using an old work appleID I no longer can access. I have hundreds of apps on the iPad so going through my "purchased" history trying to figure out which ones aren't associated with my current apple ID really isn't a realistic option, now that Apple took app management out of iTunes. When the password pop-ups appear there's NO information telling me what app is requesting it.

Everything has updated so no updates are hung waiting on password. The Apple support person that helped me in chat tonight said there was no solution other than delete all apps and re download them all. Since apparently now that app management was removed from iTunes there's no way to even get a list of apps installed by date, nevermind a way to see what ID is associated with each app. And no easy way to downgrade to older versions of iTunes.

This problem seems to be a universal one that no one has any idea how to fix and you are not providing any solution. Both of my computers constantly ask me to sign in to the iTunes store. The request cannot be avoided by canceling, and I cannot log in because I have no idea what old identity or password is required. Changing my "universal" Apple password is also useless. The only thing I can do is drag the popup to the bottom of the page so it takes up as little screen space as possible. This problem is probably affecting millions of people. Only Apple can fix this mess. Asking everyone to remove all apps and re-install them is not a solution.

Have you checked the App Store updates tab to see the apps that need updating? At least one of those is the one requiring the other password.