Please see the following:

Security and passwords settings
Usernames and passwords are not saved
Password manager - Remember, delete and change saved passwords in Firefox
Use a Primary Password to protect stored logins and passwords

Do you mean that they show up in the Password Manager?

• Edit > Preferences > Security: Passwords: "Saved Passwords" > "Show Passwords"

There is a difference between remembering the name and password in the Password Manager and a "remember me" check box on a web page to log you in automatically.
The latter usually involves the creation of a special "remember me" cookie that is stored on your computer and that is send to the server.
If such a "remember me" cookie is present then the server will remember you, so you need clear the cookies from those websites to make them forget you.

• http://kb.mozillazine.org/Cookies
• http://kb.mozillazine.org/Password_Manager

This did not help me. No passwords are showing up in my "saved passwords" file, yet it is still logging me into websites without even prompting me for my login information. That is a problem and I want to know how to correct it; I am not comfortable with my login information being stored against my wishes.

https://support.mozilla.org/en-US/kb/usernames-and-passwords-are-not-saved#w_passwords-being-automatically-cleared i think it will help u

Did you try to clear the cookies from those websites?

• Firefox/Tools > Options > Privacy > Cookies: "Show Cookies"
• http://kb.mozillazine.org/Cookies
• https://support.mozilla.org/kb/Deleting+cookies

You can remove all data stored in Firefox from a specific domain via "Forget About This Site" in the right-click context menu of an history entry ("History > Show All History" or "View > Sidebar > History") or via the about:permissions page.

Using "Forget About This Site" will remove all data stored in Firefox from that domain like bookmarks, cookies, passwords, cache, history, and exceptions, so be cautious and if you have a password or other data from that domain that you do not want to lose then make a note of those passwords and bookmarks.

You can't recover from this 'forget' unless you have a backup of the affected files.

It doesn't have any lasting effect, so if you revisit such a 'forgotten' website then data from that website will be saved once again.