I'm assuming someone you know is attempting to obtain a security clearance, and has been told that they are (or will be) ineligible unless they renounce their Italian citizenship?
I would think that they would need to contact an Italian embassy/consulate in order to officially renounce. They would then probably be required to provide some sort of official documentation that the process had been completed in order to be reconsidered for the security clearance. This is probably done by making some sort of declaration in front of an official at an Italian embassy or consulate, or maybe by submitting something in writing.
I am not certain that this is the actual procedure to follow, but it does deal with security clearances and renouncing citizenship (found on another site dealing with the same topic):
(5) Dual citizenship. Applicants who hold dual citizenship may not be eligible for a security clearance. Those who cannot receive a security clearance cannot hold an AR appointment. Therefore all applicants who hold dual citizenship must be willing to renounce their non- U.S. citizenship. U.S. citizens with dual citizenship and/or foreign passport must provide documentation renouncing foreign citizenship and relinquishing the foreign passport as an appointment eligibility requirement. They are ineligible for HPSP entry until this requirement is met. Procedures for renouncing non-U.S. citizenship: (a) Applicant signs a notarized letter addressed to appropriate embassy relinquishing citizenship and enclosing passport. (b) Applicant provides HCR copy of the notarized letter to embassy and Federal Express receipt or proof from embassy that passport was relinquished to embassy. (c) HCR sends copy of Federal Express receipt and notarized letter or proof from embassy that passport has been relinquished to embassy with EPSQ in HPSP application. (d) HCR sends copy of Federal Express receipt and notarized letter to embassy with EPSQ.USAREC Regulation 601-105 Health Professions Scholarship Program Instruction Handbook on Applicant and Selectee Processing"