If my parents have a living trust and own a home and my dad passed away, does the law require his name to be removed from trust and deed?

No action need be taken.  The trust owns the property and a beneficiary's death does not change title or ownership of trust held property.

The trust is still valid but the point is to have both parents on the living trust and then it becomes irrevocable when one passes. He doesn't need to be removed from the trust and deed, it simply passes outside of probate to your mother. If your mom would like to ensure the title is updated, make sure she finds out if she simply needs to include the death certificate because ultimately she wants to prevent from having to pay transfer taxes. In essence, she can keep the title as is and everything would be addressed if she sells the property or quitclaims to someone else. If the home is in the trust name, no title needs to be changed because the trust is still a valid document.