Under North Carolina law, does a quit claim deed relieve you of financial responsibility for the property?

I'm not sure I understand your question. I think you're asking if giving someone a quit claim deed relieves you from the obligation to repay the mortgage loan. If that's your question then the answer is no. The only thing that will relieve you from paying back the mortgage loan is paying off the loan in full or having the new owner refinance the loan in their own name.

If you are talking about real estate taxes, then giving someone a quit claim deed means that you are no longer responsible for paying the real estate taxes.