Is the chief law enforcement officer required to sign the law enforcement certification on ATF Form 1 or ATF Form 4?

No. Effective July 13, 2016, the final rule “Machineguns, Destructive Devices and Certain Other Firearms; Background Checks for Responsible Persons of a Trust or Legal Entity With Respect to Making or Transferring a Firearm” amends the regulations to eliminate the requirement for a certification signed by a CLEO and instead adds a CLEO notification requirement.  Prior to submission of the application to the Director, all applicants/transferees and responsible persons shall forward a completed copy of ATF Form 1 or ATF Form 4, or a completed copy of Form 5320.23, respectively, to the chief law enforcement officer of the locality in which the applicant or responsible person is located.  The chief law enforcement officer is the local chief of police, county sheriff, head of the State police, State or local district attorney or prosecutor.  If the transferee is not a licensed manufacturer, importer, or dealer qualified under this part and is a partnership, company, association, or corporation, for purposes of this section, it is considered located at its principal office or principal place of business; if the applicant/transferee is not a licensed manufacturer, importer, or dealer qualified under this part and is a trust, for purposes of this section, it is considered located at the primary location at which the firearm will be maintained.

Source ATF: https://www.atf.gov/firearms/qa/chief-law-enforcement-officer-required-sign-law-enforcement-certification-atf-form-1-or