Well, F3 syncs all global variables with the framework itself, so calling $f3->set('SESSION.something', 'value') is the same as $_SESSION['something'] = 'value' and vice-versa. The same goes for cookies, get and post.
The JAR array is basically just a shortcut for "session_set_cookie_params()" and in it you can setup the cookie's domain, lifetime, path, httponly, etc. That alone will not allow you to do what you want.
Start by creating a login with "Remember Me" functionality. This is basically just a cookie you set on the client and then evaluate it when the client returns in order to see if you can log him in automatically or not.
Then, when evaluating the "remember me" cookie, you'll have to write some custom code to compare a (for example) "last_active" date with the current date. If it's longer than X, then destroy the stored session and redirect to login. If not, then just resume your stored session. Your app should take care of updating the "last_active" record on every request in order to keep it updated. (unless you only want to take into account the time since last login, which I doubt it because the visitor can be logged in for days)
Don't focus on using F3 for this since it doesn't have that functionality baked in. Sure, you can use F3 to set and get cookies, sessions etc, but the rest of the checks and behavior you'll have to code in plain php on your auth controller/method/helper.