These are both yes-no questions, but they do not ask for the same kind of answers. The affirmative version, "Are you tired of eating out?" is neutral as to its expectation: the speaker truly does not know whether the answer will be yes or no. It is neutral in expectation: 50-50.
Alicia: I'm enjoying these days with you in this beautiful city. I've loved all the restaurants we've been to.
Megan: Are you tired of eating out yet? If not, we can go tonight to a good seafood restaurant I know of on the waterfront.
Megan doesn't know whether Alicia is or is not tired of eating out.
In contrast, the negative question, "Aren't you tired of eating out?" is not neutral. It is prompted by evidence that suggests that the other person is--or should be--tired of eating out. Look at this exchange:
Melinda: Let's go somewhere new for dinner, OK?
Jason: Again? We haven't had dinner at home for a week. Aren't you tired of eating out? I am.
Here Jason had expected that Melinda would be tired of eating out. He is surprised that she seems not to be tired of eating out, because she is suggesting yet another dinner away from home.
Negative questions have other roles as well. A negative question may have a negative expectation, as in
Alan: I'm going to postpone my singing audition until next month.
Terry: Why? Won't you be ready this month? (Is it the case that you won't be ready this month?)
Alan: No, I don't feel ready at all! I need more time to practice.
Negative questions can also serve a "reminding" function, as in
Come on, you know you can tell me how you really feel about him. Aren't I your best friend?
Finally, negative questions are used as exclamations, but only with adjectives or adverbs that express an opinion:
Isn't she lovely in that bridal gown?
Wasn't that movie horrible?
Didn't he play beautifully?
The "exclamation" kind of negative question is often written with an exclamation point:
Isn't she lovely in that bridal gown!
Wasn't that movie horrible!
Didn't he play beautifully!
Betty Azar's Understanding and Using English Grammar* has a short section on negative questions on page A (Appendix) 13 that discusses some of these ideas.
*Third Edition (Longman, 2002)