That would probably depend on where you're located. There certainly aren't any global laws that would cover that sort of thing. You'd probably be best off consulting a local attorney. You won't get credible legal advice in a forum.

You might want to punch up your TOS to either a) have no responsibility or b) obtain copyright. If you opt for b) you will have to be proactive regarding posts and/or limit what is posted.

That said, get a lawyer. Yes, local will be the big problem, but some issues may span international. That's why you need a lawyer!

You won't get credible legal advice in a forum.

Why ever not?

I have had credible legal advice on forums before, this included. I'm not suggesting that it should do away with the need for a lawyer but there is often good advice available based on the massive amount of experience in here.

But it has been discussed many times here [google.com].

When you ask for legal advice in a forum you get what you pay for... ;)

Revisiting this topic since it was updated I do have some additional comments as regards the OP's original question:

forum owner is NOT responsible for his visitors' copyright violations or illegal activity occurring on the forum.

There is tort law which DOES hold the publisher liable. Look at several cases regarding newspapers publishing articles defamatory of others, etc. Additionally, IF illegal activity occurs on the premises (ie, theft, planning a crime, a whole host of others, etc.) and the owner does nothing, the owner can be held liable. Free speech does have limits (You can't yell "Fire!" in a crowded theater, for example). And if you have knowledge of a crime you are required to report it. Failure to do so makes you party to the crime.

Thus, if the actions of the forum members rises to the level of illegal activities, the owner should take immediate action, or will be held liable for damages.

..and this law applies exactly to which part of the planet?

What further complicates the law is figuring out which law is relevant. The site owner might live in France, host the website in the United States, have problematic posts on it from users in Nigeria, have moderation done by a volunteer in Japan and be receiving complaints from someone in Brazil.

Do bear in mind that many types of civil action result in case law that does not produce a legal requirement. When a court awards damages, that does not necessarily mean that for someone else to do the same thing is illegal, it often just means that they too should expect to compensate the injured party in much the same way.