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?