Restricting Entrance and Strengthening the Requirements on Import Controls for Trafficking of Illicit Drugs Act or the RESTRICT Illicit Drugs Act This bill requires the Food and Drug Administration (FDA) to transfer any packages of illegally imported drugs it receives to the U.S. Customs and Border Protection. The FDA may (1) debar a person from importing drugs into the United States if they have been convicted of a drug import related felony, and (2) refuse admission of an article imported by someone who is debarred, except that a counterfeit drug must be refused.