I once worked for a business that got a DMCA request for taking down a legitimate site, from a law firm acting on the site owner's behalf. They refused to listen to reason and didn't stop until we got the site owner to call their own lawyers and tell them back off. Luckily, we were outside the jurisdiction of the DMCA so we had the option of simply saying no.

I can't imagine it was the only time such a thing happened, and that alone is an argument for simply ignoring those take-down requests if possible.