Islam does not forbid organ donations.

Submitted by natasha (United Kingdom), Jul 14, 2009 at 20:17

Blood donation is considered an extremely meritorious and rewarding act of charity in Islam; since Islam exhorts us to be charitable to all of God's creation, we cannot think of any charity greater than gift of life.

"Whosoever helps another will be granted help from Allah."

Prophet Muhammed (pbuh)

Muslim scholars of the most prestigious academies are unanimous in declaring that organ donation is an act of merit and in certain circumstances can be an obligation.
These institutes all call upon Muslims to donate organs for transplantation:

• the Shariah Academy of the Organisation of Islamic
  Conference (representing all Muslim countries)
• the Grand Ulema Council of Saudi Arabia.
• the Iranian Religious Authority
• the Al-Azhar Academy of Egypt

Islam and organ donation
One of the basic aims of the Muslim faith is the saving of life. This is a fundamental aim of the Shariah and Allah greatly rewards those who save others from death.

Violating the human body, whether living or dead, is normally forbidden in Islam. The Shariah, however, waives this prohibition in a number of instances: firstly in cases of necessity; and secondly in saving another person's life. It is this Islamic legal maxim al-darurat tubih al-mahzurat (necessities overrule prohibition) that has great relevance to organ donation.

"Whosoever saves the life of one person it would be as if he saved the life of all mankind."

Holy Qur'an, chapter 5 vs. 32

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