Should sections containing material that violates Islamic law, such as musical greeting cards, be removed from hosted websites, or is offering advice sufficient? And if the website owner refuses to remove them, must they be removed despite the potential damage to the hosting service's reputation?

You did well to stipulate that nothing violating Islamic law should be included, such as music, images of women, or songs. Even if you hadn't stipulated that, they are not permitted to include forbidden things, because the Prophet (peace be upon him) said: "Whatever I have forbidden for you, avoid it." And it is not permissible for you to condone anything forbidden or cooperate in it, for Allah the Almighty says: "And cooperate not in sin and aggression." If one of them includes something forbidden, it is obligatory to advise him to remove it, otherwise you have the right to remove it and not renew the contract with him. And do not fear damage to your reputation, for Allah suffices whoever seeks His pleasure by incurring people's displeasure.