I filed an LLC in North Carolina but live overseas
Online businesses are taxed just like any other business for income tax purposes and as a US citizen you are subject to tax on worldwide income. If you are living abroad, you may qualify for an earned income exclusion for wages you earn overseas, but profits from your US business would still be subject to state and federal income taxes in the US.
You also may be thinking about sales tax for the state of North Carolina. Online businesses are subject to sales tax just like any other business, except with an online business you are much more likely to be shipping to an out of state address than you would with a business selling from a physical location. First sales tax would depend on what it is you are selling. Generally any type of tangible good would be subject to sales tax if you deliver the goods in North Carolina. Assuming you don't have nexus in any other state, then you would only need to collect sales tax on sales to North Carolina customers. There are some services which are also subject to sales tax in North Carolina, so you might want to check with the North Carolina Department of Revenue http://www.dor.state.nc.us/index.html for the specific items you sell.