Even if you live in AK, you are subject to NC state taxes for any income you earn in NC (e.g. rental income).
When a non-resident earns money in a state, that income is subject to tax by that state. Depending on how much money you made in the "non-resident " state, you may not have a tax liability. Also, each state has filing thresholds - if you don't make more than a certain amount there, you don't have to file a non-resident return.
For NC, if you are a nonresident, you must file if you received income for the taxable year from North Carolina sources that was attributable to ownership of any interest in real or tangible personal property in the State or derived from a business, trade, profession, or occupation carried on in North Carolina, or is derived from gambling activities in this State and your total income from all sources both inside and outside the State, exceeds the amount for your filing status shown in the Filing Requirements Chart.
This webpage of the NC DOR lists the income filing requirements.
http://www.dor.state.nc.us/taxes/individual/filing.html
