No. Your father is not responsible for the debt, unless he co-signed as well. The loan should not show up on his credit report or impact his credit rating. The bank has no recourse on your father, although they could have recourse on your mother's estate if she had significant assets.
A lot of things must happen before a creditor can get a judgment or seize assets.
I can't guarantee that the creditor won't be able to seize any assets that were your mother's. However, anything that is in your father's name cannot be touched - and his credit rating will not be impacted.
If you are concerned about what will happen if you default, and your father is concerned that you will have to pony up some of your late mother's assets, have him give the lawyer that drew up the living trust a ring and he should be able to give you a more definite answer since he knows more about how your assets are structured.