I live in NH and file a non-resident MA tax return. However, my MA return shows taxes owed on my interest and dividends earned while living in NH. Why?

Yes, you are correct, you do not owe taxes on interest/dividends in Massachusetts.

While you are in the Massachusetts state return input, continue through the interview and on the summary screen ,chose Adjustments, and continue until you arrive at the screen Interest and Dividends.  Here, select Yes to make adjustments to the amounts that auto-computed.