My non-profit organization hired a consultant to do work for us in a foreign country. The amount was over $600. If the person is a citizen of another country (not the U.S.) then do we need to file a 1099-misc? This person does not pay taxes in the US.

If the consultant is not a US citizen or resident and did not work in US, you do not file 1099-misc. Income is not taxable in US.
Get form W-8BEN from the consultant.