Bring Animals Relief and Kibble Act of 2023 or the BARK Act of 2023 This bill provides civil and criminal liability protections for the donation of pet food and pet supplies under the Bill Emerson Good Samaritan Food Donation Act. (The act shields persons that donate and distribute food to nonprofit organizations from liability for injuries caused by those donated products, with exceptions.) Specifically, the bill limits liability for persons or gleaners arising from the nature, age, packaging, or condition of food or pet supplies that are donated in good faith by the person or gleaner to a nonprofit organization, state, or local government for direct distribution to pets, service animals, or emotional support animals. Gleaner means a person who harvests for free distribution to the needy, or for donation to a nonprofit organization, an agricultural crop that has been donated by the owner. Further, the bill limits liability for a nonprofit organization, state, or local government arising from donated items received from a person or gleaner in good faith for direct distribution to pets, service animals, or emotional support animals. The bill also specifies that the existing exception that permits a person, gleaner, or nonprofit organization to be held liable for acts or omissions that constitute gross negligence or intentional misconduct applies to the liability protections established by this bill.