You can fix the issue of Firefox keep asking master password by following the steps given below:

• Open the Mozilla Firefox and click on the Menu button and then select Options.
• Click on the Privacy and Security > Logins and Passwords
• Remove the check mark that is next to Use a Primary Password by entering your current password.
• Click Remove to remove it and then exit from the Firefox.
• Now again open the Firefox and go to Login Password option for reactivating the primary password and then exit from the browser.

You can even call Mozilla Firefox customer support to take the help of the experts as they are certified in solving the issue faced by you.