Firefox.
Click Tools > Clear Recent History > Expand Details and see if Active Logins is checked. If so, uncheck it. That setting would basically log you out every time you left. You also could be set to private browsing. To check that look on your address bar and see if there is something that looks like a mask. Or you can click tools and see if it says ‘stop private browsing’ (click that).
If not one of those, I’m not sure.