No such behaviour here.
There is no such behaviour with Firefox, KeePass and the Kee plugin.
I guess you have installed some other KeePass plugins.
I use Kee and Firefox and my KeePass database is nomaly locked.
There are never any pop-ups even if I enter a web page
for which KeePass and Kee plugin provide login data.
I have to unlock KeePass and only then Kee will provide the login data.
If I don’t unlock KeePass the Kee Icon in Firefox stays red, thats all what happens.
No pop-ups or anything else.