Nothing seems to happen or change if I just click away from a site.
There are two ways that web masters store your credentials, a session or a cookie. A session will be deactivated once you leave the website (i.e go from website.com to website2.com) or after it expires. A cookie is similar but it wont be deactivated if they leave the website, it will only be deactivated if they delete the cookie or it expires.
So by not logging off (even though its convenient) posses a security issue. That's the only consequence I could think of.
Eventually the session on the server expires and the credentials you used to log in originally expire with it. Of course if those credentials are cached on you local computer they could automatically log you back in but on the server you are done.