Integrity Restoration Strategy Act of 2013 or the IRS Act of 2013 - Provides that if the Internal Revenue Service (IRS) receives an application for recognition of an organizations's tax-exempt status and requests additional information to process such application, the IRS shall: (1) include a due date in the request for providing such additional information (not less than 30 days), (2) conclude that an application is abandoned if no response if received by the due date, and (3) provide a written substantive response to the organization's response to the request for additional information not later than the number of days given to the organization to respond to the IRS request. Deems an application approved if the IRS fails to provide a response within the required time period, unless the IRS demonstrates to a court that the organization does not qualify for tax-exempt status.

Requires the IRS Commissioner to: (1) complete implementation of all recommendations set out in the report of the Inspector General for Tax Administration of the Department of the Treasury titled "Inappropriate Criteria Were Used to Identify Tax-Exempt Applications for Review" and report to Congress; (2) establish new procedures, accountability measures, and supervisory positions to ensure that proper supervision and oversight is provided to all IRS departments; and (3) report to Congress quarterly on such procedures and the employment status of employees found responsible for targeting conservative groups.