The "stay logged in" feature keeps you logged in till you manually log out, no matter the time you spend on the forum, the times you close and re-open the browser, etc. It would only be "disrupted" if you clear your cache/cookies. If you do not check that, then you will eventually be logged out after a while, after closing the browser and re-opening the forum, etc.
Not checking it is useful if you're sharing a computer or you're at a public place, as then you have less to worry about if you need to leave in a rush --- manually logging out is not that important in that case.
The feature doesn't have anything related with your time on the forum being "logged", it's about keeping your cookies active so you don't have to repeatedly login with each visit.
