[The Ruling on Transplanting an Organ from a Deceased Person]

Answer – and with Allah is success: There are legal principles agreed upon among the scholars of the Muslims, namely:
1. That the Muslim, alive or dead, has an inviolability that must be preserved and respected.
2. That severing a limb from among the limbs of a Muslim is not made lawful by (his) permission; and likewise causing him pain, killing him, humiliating him, and degrading him.
3. It is permissible to sever a limb of a Muslim and cause him pain when there is necessity for that – such as when the limb is afflicted with cancer.
4. Necessity makes prohibited things permissible, to the extent that the necessity is removed.
Upon consideration of what has preceded, the permissibility (of transplanting) becomes apparent.
The proof of permissibility, which indicates that transplantation is permissible, is necessity. The scholars of the madhhab have said, as in al-Azhār and its commentary: It is permitted under coercion to commit any prohibited act, except causing pain to a human being, killing him, committing fornication, and the like.
Their words thus indicate the permissibility, under coercion, of what is other than those; and among that is transplanting an organ from the dead by carrying out a surgical operation.
[The Ruling on Causing Pain to a Human Being]
It is permissible to cause pain to a human being:
1. Either to repel a harm – such as surgical operations to remove disease, and cauterization for healing, and the like.
2. Or to bring about a benefit – such as circumcision, bloodletting (faṣd), and cupping (ḥijāmah).

Source: Min Thimār al-ʿIlm wa al-Ḥikmah vol.2