According to the morality of the Catholic Church are there any organs that are not permitted to be offered in organ transplants?

Yes, the brain and the gonads are not permitted even if the donor is dead.

The general principle regarding the donation of organs is articulated in section 2296 of the Catechism of the Catholic Church.

Organ transplants are in conformity with the moral law if the physical and psychological dangers and risks incurred by the donor are proportionate to the good sought for the recipient. Donation of organs after death is a noble and meritorious act and is to be encouraged as a manifestation of generous solidarity. It is not morally acceptable if the donor or those who legitimately speak for him have not given their explicit consent. It is furthermore morally inadmissible directly to bring about the disabling mutilation or death of a human being, even in order to delay the death of other persons.

​ Distinction is made between

• transplants from a dead body
• transplants from a living body which would cause the disabling mutilation or death of the donor
• transplants from a living body which would not cause the disabling mutilation or death of the donor.

The donation of body parts from a dead body is approved, and encouraged, but the donation must be explicit, whether by the deceased, prior to death, or by his representative afterwards.

The donation of parts from a living body which would cause the disabling mutilation or death of the donor is not permitted. A heart transplant would, of course, have such an effect.

Transplants which do not result in disabling mutilation are subject to a test of proportionality. This requires balancing the dangers and risks to the donor against the benefits to the recipient. Because the human body contains two kidneys, but can function with one, a kidney transplant would ordinarily come into this category.

The Catholic Medical Association(UK) has stated there is no objection, in principle, to transplants from animals to humans.

Pope Francis has described organ donation as a testimony of love for our neighbour.

In the case of reproductive function section 2376 of the Catechism of the Catholic church reads:

Techniques that entail the dissociation of husband and wife, by the intrusion of a person other than the couple (donation of sperm or ovum, surrogate uterus), are gravely immoral. These techniques (heterologous artificial insemination and fertilization) infringe the child's right to be born of a father and mother known to him and bound to each other by marriage. They betray the spouses' "right to become a father and a mother only through each other."

The Pontifical Council for Pastoral Assistance and Healthcare Workers , in 1995, listed the brain and the gonads as forbidden from transplantation:

The brain and the gonads may not be transplanted because they ensure personal and procreative identity respectively. These are organs which embody the characteristic uniqueness of the person which medicine is bound to protect.” PONTIFICAL COUNCIL FOR PASTORAL ASSISTANCE AND HEALTH CARE WORKERS, Charter for Health Care Workers (1995) 88.

Transplanting a brain is prohibited because personal identity is held to reside in the brain.

A gonad is either a testis in the case of a gentleman, or an ovary in the case of a lady. The prohibition here is about procreative identity. Children, it is intended, should be the genetic descendants of the married couple to whom they are born. If Mr A were to receive a transplanted testis from Mr B and then impregnate Mrs A, the resulting child, Master or Miss A, would not be the genetic child of Mr and Mrs A, but of Mr B and Mrs A. Similarly if Mrs A received an ovary from Ms C, and was impregnated by Mr A, the child would be the genetic child of Mr A and Ms C. This is the issue.

In the case of a prepubescent boy facing chemotherapy, his testis was removed and later transplanted back. This would not be a problem to the Church as it was not a transplant from another person.