In general, iPhone is known to come up with Apple Account Verification pop-ups after installation of an update; which in most cases goes away when the User verifies the Apple Account by entering the correct password linked to the Apple ID.
If the Apple Account Verification pop-up does not go away and your iPhone keeps bothering you with repeated “Apple Account Verification” pop-ups, the reason could be due to a pending App Store download, iCloud syncing issue, multiple devices using the same Apple ID, outdated credentials or payment information.
Note: In certain cases, the problem could be due to Apple ID or Apple Account being placed on hold and this can only be resolved by contacting Apple support.
Fix Repeated “Apple Account Verification” Pop-ups
If your iPhone is bothering you with repeated Apple ID or Apple Account verification pop-ups; you should be able to fix the problems using the following steps.
1. Enter Correct Password
When the Apple Account Verification pop-up appears, make sure that the Apple ID mentioned in the pop-up is linked to your device and verify your identity by entering the correct password associated with your Apple ID.
1. When the Apple Account Verification pop-up appears, tap on Settings.

2. On the next screen, make sure that you are entering the correct password and tap on Sign-In.
2. Sign Out And Sign-in to iPhone
This step can help, if the problem is being caused due to an iCloud glitch or conflicting Apple IDs.
1. Go to Settings > tap on your Apple ID Name.
2. On the next screen, make sure that your Apple ID (Email Address) is correct, scroll down to bottom of the screen and tap on the Sign Out option.

3. Follow the instructions to Sign-out from iPhone.
After you are completely signed-out, wait for 30 seconds and restart your device by pressing the side or power button.
3. Check for Updates
The reason for the pop-up could be due to an update or app download waiting to be installed on your device.
1. Go to Settings > General > Software Update.

2. On the next screen, wait for iPhone to search for updates
3. Follow the instructions to download and install the update (if avilable).
4. Check Your Payment Information
Make sure that the credit card or other form of payment information linked to your Apple ID is correct.
1. Go to Settings > tap on your Apple ID Name.

2. On Apple ID screen, tap on Payment & Shipping > If prompted, enter your Apple ID Password.
3. On the next screen, tap on your Credit Card or the listed Payment Method and make sure that the credit card or the listed payment method has not expired.
5. Force Restart iPhone
Force Restarting the device has been listed as a potential fix for this issue on Apple support forums and this can help in fixing the issue by clearing out temporary bugs which might be responsible for the problem.
The Force restarting of iPhone can be achieved by pressing and releasing Volume up, Volume down and Power button in a sequence, as provided in this guide: How to Hard Reset iPhone (All Models).
6. Login to Apple ID Page
If the problem is not yet fixed, login to your Apple Account on a MacBook or Windows computer and see if there is a need to Reset Apple ID Password or update/re-confirm answers to security questions.
1. Using a Mac or PC go to account.apple.com.
2. Enter your Apple ID and Password to login to your Apple Account.
Once, you are logged-in, you may see a message requesting you to update your Apple ID password or confirm your answers to Security questions.
3. Follow the prompts to Change/Update your Apple ID password or Reset Security Questions.
After you are done with this, you may still find iPhone coming up with “Apple Account Verification” pop-up, the first time you go back to your device and also while trying to use iMessage.
This happens, due to your Apple ID being linked to all iCloud related services. Once, your Apple ID is verified on two or more iCloud services, the problem should go away.