The specific situation you cite:
Q: [Any question whatsoever]
A: How can you worry about that when children are starving in Africa!
is called the Fallacy of Relative Privation. See http://en.wikipedia.org/wiki/Fallacy_of_relative_privation
There are many other ways in which a question can be answered in a manner that explains nothing though. For example, "begging the question" is answering a question in a manner that assumes the thing being asked about:
Q: Why is this diamond hard and this butter soft?
A: Because the diamond is made of hard molecules, butter is made of soft molecules.
You might have learned that stuff is made of molecules, but you have learned nothing about hardness and softness from this answer. Why are hard things hard? Because hard things are made of hard things. That is really a non-answer.
There are many different ways in which a question can be non-answered; are you looking for a term for all of them, or for some specific examples?