Is it correct to ask:
What do people living in intelligent homes have not to worry about?
What do people living in intelligent homes not have to worry about?
Setting aside the issue of intelligent homes, the usual construction is what do people have to worry about or what do people not have to worry about.
Regarding your first example, it's possible to imagine a context in which somebody says:
There are so many things that I have to worry about right now.
To which someone else might respond:
What do you have not to worry about.
It's a play on words asking: What is it you have that you are not worried about?
But it's not typical.
So while both of your constructions are possible, the second is idiomatic and far more likely in most contexts.