Unless you have "Remember login" set your going to have to enter your login every time you go to your email. With the saved login you click the right email and then select the matching pswd to login. Since AOL isn't Firefox browser Mozilla has nothing to do if there are AOL issues.

how to open aol mail in new tas without signing in to aol every time

Make sure you check the "Stay signed in" box when you log in, and allow cookies in your browser. That should keep you logged in so AOL Mail opens in a new tab without asking again