I terminated my U.S. resident alien status (gave up my green card) and was told that I may still need to furnish some documents to the IRS. Can you please explain?

If you are a U.S. resident alien, the rules for filing income, estate, and gift tax returns and for paying estimated tax are generally the same whether you are in the United States or abroad. If you are a nonresident alien, you are usually subject to U.S. income tax only on U.S. source income. Under limited circumstances, certain foreign source income is subject to U.S. tax.