Automatic sign-out when you leave a webpage?

gogreen said:

Suppose I sign in to a forum or to other website for which I have a user name and password. If I leave that website without signing out, either by closing my browser or going to another website, does that website automatically sign me out?

As a rule, no.
Mostly, it hinges on the expiration of the site's data, (cookie).
Most browsers can be configured to clear data on exit.

Also, most sites that require a log-in will prompt your Browser as to whether to save the log-in or not. If a site is questionable I don't save the log-in.

gogreen said:

Suppose I sign in to a forum or to other website for which I have a user name and password. If I leave that website without signing out, either by closing my browser or going to another website, does that website automatically sign me out?

An easy way to find out if it logs you out when you leave without manually signing out just re-visit the webpage after you close the tab/browser. If it saves your data/sign in you will see you are still logged in. If it doesn't save it then you will need to sign in again.