On the machines that require the password, how frequent are you visiting that site on that machine? Is it similar to the machines that do not require the password? Perhaps the login has expired?
Is there any difference in how you have data clearing settings between all of the machines? See Settings -> Privacy & Security -> Cookies and Site Data. See also the History settings on the same Settings page. Do you have "Firefox will" set to Remember History or some other choice and is it the same for all machines? If you set it to "Use Custom Settings for History" you can then clear data on exit. Verify you haven't changed that on the "problem" machines. One of the history clearing choices is Active Logins.
Give a man a fish, and he eats for a day.
Teach a man to fish, and he eats for a lifetime.
I like poetry, long walks on the beach and poking dead things with a stick.
Please do not PM me for personal support. Keep posts here in the Forums instead and we all learn.