I have a rental property in NC. I completed the state taxes and I do not owe nor am I receiving a refund. Do I need to pay the $59 to file the returns?

It depends. The North Carolina Department of Revenue web page states Non-resident individuals are required to file a 2022 North Carolina individual income tax return if they received income for the taxable year from North Carolina sources that was:
 

(1) attributable to the ownership of any interest in real or tangible personal property in North Carolina, or 

(2) derived from a business, trade, profession, or occupation carried on in North Carolina, or 

(3) derived from gambling activities in North Carolina 

and 
whose total gross income from all sources both inside and outside of North Carolina for the taxable year exceeds the amount shown in the Filing Requirements Chart for Tax Year 2022.  

So, if your income is over the filing limit, you will need to file a North Carolina return and if it is below the filing limit, it will not be required. 
 

 [Edited 03/14/22|12:45 pm PST] to clarify filing limit