My informant started hearing this saying two years ago during his junior year of high school.  People were saying it everywhere, and he couldn’t help but say it too.  “Don’t worry about it” became part of his more used expressions.  Eventually, all of his friends started saying it too.  It became one of the phrases that they would know when to use.  They used it so frequently that they would answer any question or comment with “don’t worry about it.”
He uses the saying when, obviously, he doesn’t want someone to worry about something.  He replaced “it’s alright,” along with other phrases, with “don’t worry about it.”  He started saying it even when it didn’t really apply.  If a person asked him a question and he didn’t feel like answering, he would just reply with “don’t worry about it.”
He does not know what he would do without this saying.  He has begun to rely on it so much because it allows him to answer even when he doesn’t know what to say.  It’s also become special to him because it’s pretty much an inside joke between him and his friends.  “Don’t worry about it” is accepted as an appropriate way of responding his friends, but not so much with others.
I think that this phrase is just a way of avoiding things.  Some usages of it is fine, but when someone answers back with “don’t worry about it” when there should be an actual answer, I think that the person is just trying to avoid questions.  I think that my informant and his friends shouldn’t rely on this saying when they’re having conversations with other people.  Replying with “don’t worry about it” after every few questions can become quite frustrating, especially when you need answers.