A friend of mine runs a forum about one particular item and has ran this forum for the last few years without any issue. However the company that makes this product now wants him to censor or run his forum in a way they see more fit.
Being that his domain name has the company product in it, as an example: www.AMDathlonForum.com, would the company have any legal rights to make him change his censorship or the way he runs it because he has the company name/product in his domain name?
Could they take the domain name away from him?
Wow, that sounds tricky. If the company has a trademark (not just copyright), they would have a little more power to sue for damages including getting the domain name. If I were your friend, I would try to work out something financially with the company, since there is a joint benefit.
They would have no right to dictate how he runs his site. Are you sure they've actually asked to do this? How did they word the approach?
As for the domain name, yes it's possible they could pull the trademark card on it, but unless they've mentioned it themselves, it may not be something to worry about until it actually happens.
I would first try to work out the reasoning behind their approach. I would assume the forum is great advertising for their product, plus also provides free customer support for them, so why are they suddenly taking such a dim view of things? Are there particular posts they want removing? Is it purely because they don't want people posting negatives about the product?