Immigrant Fairness Restoration Act of 2001 - Amends the Immigration and Nationality Act to revise specified restrictions respecting: (1) cancellation of removal; (2) deportability and admissibility; (3) judicial review; (4) right to counsel; (5) detention; (6) applicants for admission; (7) removal of aliens who are not permanent residents and who have no relief from removal; (8) reopening of certain proceedings and parole of certain aliens; and (9) aliens who have made false citizenship claims or who have unlawfully voted.