I had no earned income this year but paid student loan interest. Do I have to file taxes?

If you had no income of any kind during 2016 there is no reason to file a tax return.  The student loan interest paid is reported as an Adjustment to Income, but since you would have no income to report the interest paid would not be relevant.

Are you married?

How did you pay the student loan interest if you had no income?   You might still need to file a tax return if you had sufficient taxable unearned income.

I'm in a similar scenario. I had no income, but I paid my student loan interest from my savings.

@ejohnson923 - you made a comment to a 2 year old post that makes no sense.  If you have a question then ask a new question and provide details.