This feature never works for me and has not over the last 14 years. How about removing the option from the sign-in page and do everyone a favor?
It works, but sometimes it gives the appearance of not working.
I always use the "sign in for 2 weeks" option. When I first open my browser(s) in the morning, The "Sign In" tab/button shows at the top right. Without signing in, I enter my search parameters, and the window will self-refresh, and the Sign In button is replaced with my username and personal icon.
It has always worked, but it has never done quite what it says it does. It doesn't prevent you from getting timed out (signed out due to inactivity) -- i.e., it doesn't actually "keep you signed in". What it does do, as Áine points out, is sign you back in without you needing to re-supply your credentials. If you've been timed out, but the 2-week cookie is still active, then you will be automatically and silently signed back in if you do anything that requires you to be signed in, such as going to any Family Tree page. Therefore, you should only need to supply your login credentials once in every two-week period. (And you can teach your browser to supply them for you.)
I believe there are browser security settings that can (silently) disable the option, but that's not a topic I know all that much about.