"Thanks for watching Lord Fluffington! You don't have to cook him gourmet meals; the canned food in the pantry is perfectly fine. However, you mustn't look him directly in the eyes after midnight, or he will hold a grudge for a week."

Don't have to shows that something isn't necessary. You aren't required to cook gourmet meals, because canned food works just fine.

Mustn't expresses a strict rule or warning. Looking the cat in the eyes after midnight is strictly forbidden unless you want him to be angry!