I think everyone gets this problem at some point on their board. A member is banned, or gets warned and doesn't like it, and starts throwing all kinds of threats at you, things like they will prosecute you unless you remove their account and posts.

Don't listen to them.

I'm here to tell you that. According to basic terms and conditions which are there by default in phpBB, MyBB, IPB and vBulletin, anything posted on your forums is solely your property. This means if a user posts something on your forum which belongs to them, they are also allowing you legal permissions to use it and redistribute it. That is there by law and unless you remove it from your T&C, will always protect you.

Next time you get a member asking for their posts and account to be deleted, cite the following to them: "Under the copyright act of 1990, any article, post, or image you use on this board is respectively our property, you cannot prosecute us to get us to remove the content. Although it is indeed copyrighted to yourself, when you posted it on the board you gave us, (Forum name), full permission to use it with or without your consent."

Then they either feed that to their lawyer, who confirms that it's right and closes the case, or they give up. Most of the time you just have to say that and then they'll stop harassing you. It's easy enough. Don't fall for their silly tricks and long phrases and words which you might not understand, and don't feel intimidated by them - You're the person who's in the right!