It depends on the site. In most cases staying logged in to a forum on your own computer is not going to cause any problems and is convenient. Even if it did lead to someone breaking into your account the worst that can happen is that they post forum comments as you. That doesn’t make it all that useful for someone to break into your account anyway.
For other types of account where you need to login, actually logging out is far more important (eg. you wouldn’t want to stay logged into your internet banking as you want as many layers of security to protect that as you can.
Of course not everyone knows to distinguish between sites based on the risks and so encouraging them to stay logged in to a forum may result in them doing the same on other sites increasing the chance of a successful attack on an account of theirs where they should be logging out.