My sister says I ask too many questions, such as "What have you been up to lately?" She is the only person who says that.
What do you call a person who doesn't like questions being asked?
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Another good word would be reticent: inclined to be silent or uncommunicative in speech.
There may be different reasons why questions about oneself are unwanted.
shy people feel uncomfortable being in center of others attention and that includes being asked a lot of questions.
secretive people either have something to hide, or just don't like ttalking about themselves much.
Same goes for silent in the meaning Not inclined to speak; not talkative.
In answer to your question "What do you call a person who doesn't like questions being asked?", I think of such a person as an authoritarian, self-important, fascist, close-minded, stupid, supercilious, cross-grained, arrogant and imperious fool, dunderhead, or dictator.
But perhaps that's not what you want to know, and perhaps meant to ask, "How should [Maureen] refer to this question-averse aspect of her sister?", which of course I don't know. However, if previously-suggested shy, silent, private and secretive don't apply, also consider veiled, introverted, taciturn, censorial, no-nonsense, thin-skinned, touchy and untalkative.
The ODO proposes this rather good example sentence:
‘The question took him by surprise, and his eyes grew guarded and cautious, his own grip tightening as well.’
You might say that she answers your questions guardedly, too.
There isn't a correct answer to your question. Whatever adjective you choose largely depends on your sister's personality traits and life experiences. That said, one might say your sister is shy, introverted, reserved, reticent, secretive, insecure, taciturn or unsociable. Whichever I choose would be a real guessing game, though.
Based on analysis of Gardner's multiple intelligences, the faculties of the mind and the characteristics suggestive of the learning disabled: adding inattentiveness or inadvertance, bodily or mindfully impaired and/or lost, visually or aurally illiterate, unfocused, corrupt, unreasonable to be a fool! However, the summarized fool is based off memory and attention issues. He may not like other conversational circumstances. If he doesn't like conversational circumstances entirely he is sociolinguistically (conversationally) incompetent, Noam Chomsky might say. The person who doesn't like answering questions in the worst case would be a fool. This question has helped me to see that incompetence, impairment, mindlessness, illiteracy and foolishness can be a rolling problem. It's only rolling if the worst case scenario fool has memory and attention problems, and the rest aforementioned. Thanks for helping me study for the Praxis PLT with this archetype!