Election Integrity Mail Reform Act of 2023 This bill establishes certain requirements related to election mail. Election mail means any item mailed to or from an individual for purposes of the individual’s participation in an election for public office, including balloting materials, voter registration cards, absentee ballot applications, polling place notifications, and photographic voter identification materials. Specifically, the bill requires the U.S. Postal Service (USPS) to give priority to the processing and delivery of election mail. The USPS must (1) deliver any election mail regardless of the amount of postage paid, (2) process and clear election mail from any postal facility each day (to the greatest extent practicable), and (3) expeditiously carry and deliver election mail. Additionally, the bill directs the USPS to place a marking or notice on election mail to indicate that the piece of mail is election mail. The bill allows a national, state, or local committee of a political party that is eligible to mail at the nonprofit rate to conduct a cooperative mailing at that nonprofit rate with a candidate, a candidate’s committee, or another committee of a political party. The national, state, or local committee of a political party may seek reimbursement for the costs of such mailing. The bill also imposes certain criminal penalties for (1) forging or counterfeiting any postmarking stamp or impression thereof with respect to a mailed ballot for a federal, state, or local election; or (2) falsifying the date on which a postmark is applied to a mailed ballot.