Problem: Your iPhone keeps asking you to enter your voicemail password, but you don't remember it.
Solution: Reset your password using AT&T's automated system with the instructions below.
From AT&T's website...
Reset voicemail password with AT&T's automated system
Call 611 from your iPhone,
Press 1 to confirm your phone number,
Press 5 to go to voice mail help,
Press 1 to reset your password,
Enter the billing zip code for the account.
AT&T will send you a text message with a temporary password
Remember or copy the temporary password
Go to Settings > Phone > Change Voicemail Password
Change your password