Does anyone know what to do if an employer is a non-profit org and did not take out federal taxes but took out for state and FICA. Should they file only state or both?

Did you file a W-4 with your employer? Did you say on the W-4 that you would not be liable to pay federal income taxes? If so, why?

If you did not claim an exception to owing federal income taxes, then ask your employer why they didn't withhold anything for your federal income taxes.

The fact that your employer is a non-profit does not excuse you from owing federal income taxes. What do you do for this employer?

Without knowing more information about you, I would say that generally you are liable to file both federal and state returns, but this depends on what kind of work you do, what your filing status is, what your adjusted gross income is, and so on. We simply don't have enough information at this point to give you a definitive answer.