Consensual Donation and Research Integrity Act of 2023 This bill requires certain entities that acquire human bodies and sell for profit a whole human body or human body part to register with the Department of Health and Human Services. Registered entities must comply with record-keeping, labeling, packaging, and disposition requirements. The registration and related requirements do not apply to the Organ Transplantation Network (a public-private partnership that links the professionals involved in the U.S. donation and transplantation system) or its members; the preparation, transportation, and final disposition of human bodies or body parts by funeral service professionals; and schools of medicine, dentistry, and mortuary science and other research and training entities that do not sell whole human bodies or body parts for profit.