This usually happens after an update or may be after you've restored your iPhone. Even otherwise, let's check this out: your iPhone keeps asking for the Apple ID password. There's nothing downloading, there are no wrong passwords, there's no problem with your account. And yet you keep getting this pop-up which makes life unbearable with the iPhone.
It's usually a case of failed downloads which aren't directly visible on the homescreen. Sometimes it's a problem with your account settings: either in iCloud or iMessage/FaceTime or in the App Store.
Let's try to fix this as smoothly as possible. If it doesn't look smooth, my apologies.
How Do I Fix This Issue: iPhone Keeps Asking for Apple ID Password!
The first thing you do in such a circumstance is reboot your iPhone by pressing home button and sleep/wake button simultaneously.
#2. Reset Settings
You can also try resetting the iPhone without erasing the data. (Settings → General → Reset → Reset Settings.)
#3. Check Updates/Purchased Apps/App Store/iTunes Store
The first thing you do is open the App Store and check your Purchased Apps history. The Updates section shows you apps that are ready for updates but above it, there's the purchased apps section. Check to make sure there's nothing hanging in a failed download or something. These may/may not show up on your homescreen so this is the only way to check if there's an app hanging midway in a download that's failing.
Next up, open your account settings in the iTunes & App Store (Settings → iTunes & App Store) and try logging out of your Apple ID and then log in again. This might solve your problem or give you a hint about where the problem could be.
When you do this, if you find that your are unable to login to your Apple ID, this means there's something wrong with your Apple ID or the password. Try a password reset from your Mac/PC and then come back to your iPhone/iPad again and login with the reset password.
Did you check your iCloud account? Is it configured correctly? Try turning off iCloud by deleting your account. (take a backup both in iCloud and over iTunes before you do this).
When you get to Settings → iCloud, tap on the Account field and erase and retype the password for your Apple ID. Click on Done afterwards and see if this fixes the issue.
If things are still the same, you're probably going to have to check out iMessage and FaceTime under Settings. Both iMessage and FaceTime use your Apple ID even if you're not using these services (but have them turned on). There might be some problem with the account information, or activation.
Try turning these off (I know it can be cranky to get them back ON but you'll have to try it!). If it works, then Voila! You've fixed the source of the issue. Re-activate iMessage and FaceTime with your correct Apple ID and password.
#5. Change Your Apple ID Temporarily
#1. Launch Settings app and then scroll down and tap on iCloud.
#2. Scroll down and tap on Sign Out and confirm it. Just in case, you have iOS 7 or earlier, you need to tap on Delete and confirm it.
#3. Next up, tap on Keep on My Device or Delete from My Device. Enter the password of your Apple ID. The data of your phone will remain in the iCloud and when you will sign in, it will be updated.
#4. Now, you need to go to My Apple ID and then Click on Manage your Apple ID. Sign in with your existing Apple ID and password.
Up next, click on Edit which is next to the Apple ID and Primary Email Address.
#5. Now, you will have to change the Apple ID to the email address which iCloud ask you to sign in. You don’t require to verify your email address.
#6. Sign out of My Apple ID.
#6. Update your iDevice to the latest version
#1. First off, backup your iPhone. Then, update it to the latest version. Settings → General → Software Update.
#2. Once your device has been updated, sign out of iCloud.
While restoring your backup, if you are asked to enter the password of your previous Apple ID, Skip it.
Once you have restored your device, sign in to your iCloud account.
#7. What if nothing works?
Are you sure iTunes Store isn't trying to download something which is on a download-loop or failing repeatedly? Check it out by heading over to the iTunes Store on your iPhone. If there are any pending tracks that are stuck, remove them, restart your iPhone and check if this fixes the problem.
Also, if this fails to solve things, you'll just have to restore your iPhone from an iCloud/iTunes backup. A clean restore usually helps reset all things and throws away the problems. You've got the data backed up anyway so if it helps, why not just do it and get rid of the issue?