At the moment it is all or nothing. You can use private window by new...private window. This should not require a password.

• https://support.mozilla.org/en-US/kb/use-master-password-protect-stored-l...
• Protect your Thunderbird passwords with a Primary Password

In Firefox for Android there is Guest Mode, if you have a sync account.

Let us know if this helps!

You would have to disable the Password Manager temporarily to prevent a Master Password pop-up if you visit a web page with a log in form and you have previously saved a name and password in the Password Manager.

• Tools > Options > Security: Passwords: "Remember passwords for sites"

Unfortunately private mode prompts master password too.

Disabling password manager is an acceptable solution.

I became to use master password because occasionally some of my passwords disappear from password manager. With mp it seems to be okay now. Could this enabling-disabling of mp cause an unexpected erasing passwords?

Toggling the "Remember passwords for sites" shouldn't cause passwords to disappear.
I use this all the time with a button on the PrefBar toolbar to prevent the MP prompt and never have lost passwords.

You can lose passwords if you make changes to names or passwords in an input form that is filled automatically by the Password Manager, so you should be careful in such a case.

You can set the signon.autofillForms pref to false on the about:config page to prevent Firefox from filling names and passwords automatically (you need to double-click an empty input field to open the drop-down list or type the first letter).
You can open the about:config page via the location bar.

• http://kb.mozillazine.org/about:config