FWIW.
Since I hit the stay logged in square ( tho I swear I logged out one time ) I've been logged in ever since ; this is nice compared to 6 or so years of logging in every time I come here & I don't get logged off while slowly typing my long rants & reply's.
In Short
Does my staying logged on 24 7 cause t-net or me any issues ?
My understanding is the forum software (vBulletin) sets a "cookie." Next time you sign in - vBulletin checks for a "stay connected cookie with your sign-in stuff" and bingo - you are already logged in. Now, if you have your browser set to prevent software from leaving cookies...
When you log in, do you right click a topic and open in a new window? Is this when you are having trouble staying logged in?
To diagnose the problem, a fully detailed description of what you were performing may be helpful.
What operating system, browser, detailed info on what action you were performing once you logged in - that sort of detail. each step by step until the error message displayed.