This suggestion was brought up in the forums for membership sites where page reloads may not be performed very often.
This could be achieved by polling the page with JavaScript, an easy thing to with the Heartbeat API. When an old user session is detected the user could be immediately logged out.
Additionally, a filter could be exposed to redirect the user to the URL of your choosing (such as the homepage), rather than destroying the session and always being sent to the login screen.
There are performance considerations to take if a feature like this were to be added, however, the plugin could simply gracefully degrade and require a page reload if the Heartbeat API is disabled - which you should be doing anyway if performance is a concern for you.