Q: Used iPhone - asking for another person's password

You may not be able to if it is activation locked. Try these articles to see if they help.

https://www.icloud.com/activationlock/

Find My iPhone Activation Lock - Apple Support

Hi

If the iPhone is still locked (via Activation Lock), ask the previous owner to follow the steps here to remove Activation Lock and their association with your device:

Turn off Find My iPhone Activation Lock - Apple Support

Otherwise, check that you are signed in to all associated services on your iPhone with your own Apple ID, by following the steps here:

Sign in with a different Apple ID on your iPhone, iPad, or iPod touch - Apple Support

Do you mean the  iTunes that is on your computer when you try to sync your device?

About encrypted backups in iTunes - Apple Support

Or do you mean the iTunes Store (when you try to purchase songs or apps)?

Sign in to the iTunes Store on your iPhone, iPad, iPod touch, Mac, PC, or Apple TV - Apple Support

Again, iTunes or iTunes Store?

You may find that your iPhone backup contained items that were purchased under someone else's Apple ID (whether via iTunes, iBooks or the App Store).

If you continue to decline the prompts to enter those passwords, your iPhone may eventually advise that it will finish syncing and permanently remove any items that relate to Apple IDs for which have not provided the passwords.

Have you tried the steps in the section "iTunes, iBooks, and App Store" in:

Sign in with a different Apple ID on your iPhone, iPad, or iPod touch - Apple Support

Another option that appears to have worked for another user is:

Go to Settings > iCloud - you may find that it reports "Restore In Progress" (or similar) > tap on "Stop restore" > back up your iPhone to iCloud > the prompts may then stop.

More information:

http://forums.macrumors.com/threads/keeps-asking-for-other-peoples-apple-id-and- password.1265322/

You're welcome!