President: Many busy executives ask me: what about the job displacement market program in the city of the future? Well, count on us to be there Jim, because, if we're lucky, tomorrow, we won't have to deal with questions like yours ever again.
In Bleach: The following exchange is lampshaded by Souken when Uryuu asks if what Ryuuken said is true. Souken explains it's Metaphorically True but Uryuu's too young to understand that Ryuuken's only given an obvious and practical answer that completely side-steps the question of "hate" entirely.
Uryuu: Father, why do you hate being a Quincy?
Ryuuken: You can't make a living from it.
In Fight Club, after the chemical burn scene, Marla asks what happened to the narrator's hand. Tyler has asked the narrator not to talk to Marla about him.
Marla: Who did that to you?
Narrator: A person.
In First Blood when Rambo is asked what he hunts with his huge knife, he responds simply, "Game."
In Dorm Life, Josh asks Danny B about his documentary:
Karen Armstrong claims that the Biblical "I am that I am" is a Non Answer.
Similarly, some view "Thou sayest it," the answer Jesus gives to Pilate's question "Art thou king of the Jews?", as a Non Answer. Does Jesus mean "Yep, you said it; you got that right," or "That's what you say; I never made that claim"? (Jesus Christ Superstar understands it in the latter sense.)
"Render unto Caesar the things that are Caesar's and unto God the things that are God's", as an answer to whether the Jewish people should pay taxes to the Romans, is specifially stated in the Bible to be a Non Answer. The questioners were trying to trap Jesus into making a statement either way, at which point they would either declare him to be a Roman sympathiser to the Jews, or declare him to be a Judean revolutionary to the Romans. Regardless, even Jesus' enemies were impressed at how he wiggled out of that rhetorical trap.
In Lest Darkness Fall, Padway dodges inquiries about his religion (a touchy subject in sixth-century Ostrogothic Italy) by saying that he's a "Congregationalist", which he describes as "the closest thing to (name of questioner's religion) in my country".
Malaclypse the Younger: Even false things are true.
Greater Poop: How can that be?
Malaclypse the Younger: I don't know man, I didn't do it
And that's just the first page
Karlsson on the Roof, who's like some kind of a modern, urban fairy looking like a smallish man with a propeller on his back, but acting childishly, will only answer to questions about his age by saying he's "at his prime".
Lampshaded several times in Babylon 5, possibly due to the many times it was played straight, what with the Vorlons, and the Minbari, and Lorien...
Ta'lon: "While every answer is a response, G'kar, not every response is an answer."
Kosh in particular did it so much that Sheridan sniped at him about it more than once.
Sherdian: You know, I really hate it when you do that.
Sir Humphrey in Yes Minister is not fond of giving straight answers. When pressed for one, he exaggerates this trope instead, talking for thirty seconds without saying anything at all.
"Well, minister, if you asked me for a straight answer, then I should say that, as far as we can see, looking at it by and large, taking one time with another, in terms of the average of departments, then in the final analysis, it is probably true to say, that at the end of the day in general terms, you would probably find, that, not to put too fine a point on it, there probably wasn't very much in it one way or the other. As far as one can see. At this stage."
In a later episode, Hawkeye himself is accused of doing this as he attempts to avoid having to tell a patient that his bloodwork revealed he has leukemia.
Real life example: when asked in one of the interviews what happened with Russian military submarine Kursk, Vladimir Putin answered "It sunk". (Which is also a grammatically incorrectly translated non-answer, as he should've said "It sank". The Russian quote "Она утонула", is grammatically correct but still both a non-answer and incredibly cynical.)
Urban Legend: When Willie Sutton was asked why he robbed banks, he reportedly replied "Because that's where the money is."
Another real life example: When famous moutaineer George Mallory was asked why he wants to climb Mount Everest he replied "Because it is there." (Mallory died on Everest in 1924. It's unknown whether he reached the summit)
xkcd hid a joke about this in the gigantic scrolling strip with a character saying that the phrase sounds better than "because I'm rich enough for my goals to be arbitrary".
Real world example: in Zen, the answer Wu or Mu, in Chinese and Japanese respectively, is a cross between this and a Mathematician's Answer. The original question asked was, "Has a dog Buddha-nature?" The answer loosely means, "Without meaning." A philosophical STFU to an Armor-Piercing Question. The Other Wiki has a better discussion for those interested.
The Shivah enables you to respond to almost every question with a "Rabbinic Answer", which is essentially just answering the question with another question. To be fair, the point of these questions seems to be to make your interrogator realize the answer on his/her own... It only works sometimes though, as most of the time people react with impatience when you try it.
Gunnerkrigg Court: Tom Siddell does this when he doesn't want to answer a fan's question.
In Family Guy, Meg comes downstairs with a dress on, in preparation for a prom. She asks Brian, "How do I look Brian?" Brian says, not wanting to either lie to her or insult her, "Ahhhhh... You sure do, Meg."
Batman: The Animated Series used it in "Night of the Ninja." When Batman tells Robin that Kyotai is "good" at the martial arts, Robin asks how good he is. Batman reiterates, "Good." (Translation: "Better than me.")
Not in-universe (though there probably are several examples to be listed) but if you browse through the Phineas and Ferbwiki you can find the following 'information' about Ferb:
In The Simpsons episode Homer vs. The Eighteenth Amendment, Homer is using hollow bowling balls to smuggle illicit beer.
Marge: Why do you have so many bowling balls?
Homer: I'm not gonna lie to you, Marge. (proceeds to get in his car and drive off)
Jack Handey played it for laughs: "When my little nephew asked if the equator was a real line around the Earth or an imaginary one, I just laughed. Laughed and laughed. I laughed because I didn't know the answer, and I hoped if I laughed long enough, he'd forget the question."
Probably happened to you at some point: "Why can't I do X?" "Because I said so."
Q:"How do I do [task]?"
Question: "What's For Dinner/What do you want for dinner?" Answer: "Food"