You may need to file a New York City tax return as a non-resident if the work you do is done remotely as a convenience rather than as a necessity for you or your employer. In other words, if you could preform the services in New York but choose not to out of convenience, then the income will be considered New York sourced income so it will be taxed there.
You will need to file a North Carolina tax return and report the income from the New York job since you are a resident of North Carolina. You will get credit in North Carolina for tax paid to New York on income also taxed in North Carolina.
If you work for more than six months for the New York company, your tax obligation will be the same as if you work for less than six months. The time period is not critical.
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