He doesn't have to mention it - This is the typical way to form a negative. He hasn't to mention it - I do not believe this is correct English, or at the very least sounds very old. There are other ways to say this, such as "He needn't mention it." There are also phrases like: "I haven't got all day." These situations are less common and less standard, but common enough, particularly the latter.

The first is a grammatically correct sentence but uncommon. People will have difficulty deciding what it might mean.