Hmmm... Fascinating question. It's very subtle and can go in two directions."Nope" is equivalent to "no". However, interpreting the subtle meaning depends on analyzing the inflection and the context. It can be a casual "no" that is used among friends, sometimes in a a humorous situation.
However, if you are getting a "nope" from someone to whom you've asked a question they don't want to answer, it is, INSTEAD, a curt reply. It sort of ends the discussion. The sound of the "p", I think, indicates a finality to the answer. (If my husband says that to me, my feelings are a bit hurt thinking he is closing down."It's all I want to give you. I don't want to expand on the topic" is then my interpretation.
It can be sarcastic, but again, it can be an attempt to be humorous. Listen for the tone, the inflection. Analyze the situation and watch the body language. You're very astute to ask this question. Maybe some one can give you some actual dialog here that would help. I'll listen this week to my conversations and see if that word comes up and I'll get back to you with specific situations :)