Why are organs not harvested from deceased cancer patients?

On of the main reasons why cancers are normally not transmissible between different people is basically the same reason as why organ transplants are difficult: histocompatibility.

Every human cell carries a set of human leukocyte antigens on its surface, and (except for identical twins) every person's cells carry a different set of antigens. Any cells that don't carry these specific antigens are treated as foreign by the immune system and rapidly attacked and destroyed.

Normally, cancer cells carry the same antigens as the host cells they arose from, and thus are mostly left alone by the immune system. However, when tissue (whether cancerous or not) is moved from one person's body to that of another, the recipient's immune system recognizes the transferred cells as foreign, since they carry the wrong antigens, and tries to destroy them. In the case of organ transplants, this leads to transplant rejection.

The main ways to avoid transplant rejection are 1) choosing donors that happen to carry a similar set of antigens, either due to a close genetic relationship or just by chance, and 2) treating the recipient with immunosuppressant drugs to prevent the immune system from attacking and destroying the foreign tissue. With appropriate choice of donor and a well chosen drug regime, the rejection can often be prevents, or at least mitigated and delayed.

But the same methods that are used and needed to prevent rejection of the transplanted organ will also, just as effectively, keep the recipient's immune system from attacking any other cells presenting the same foreign antigens — including any cancer cells that might have been inadvertently transferred along with the transplanted organ. Thus, transplanting tissue that might be infected by metastasized cancer cells inevitably risks infecting the recipient of the tissue with the same cancer as well.

Nor is this just a theoretical concern. For example, as pointed out in Chris's answer, in 2007 four women became infected with breast cancer after receiving organs from the same donor (who was not diagnosed with cancer at the time of her death, and thus qualified as an organ donor). Three of the recipients died of the cancer, while the fourth survived after the transplanted organ was removed and immunosuppressant drug treatment was stopped, allowing her immune system to fight the foreign cancer.

---

Ps. The other main reason why cancers don't usually spread from one person to another is that, unlike infectious viruses, bacteria and other parasites that have adapted to the lifestyle of a transmissible pathogen, most cancer cells simply don't have any effective mechanism of getting out of their current host and into a new one. That's why even identical twins aren't generally at significant risk of contracting cancer from one another through casual contact, even though their cells bear the same antigens.

Even so, a few lineages of cancer cells have managed to overcome both of these obstacles and to survive their original hosts as transmissible diseases. While no such clonally transmissible cancers are known to naturally occur in humans (the few recorded cases of human cancer transmission arising from organ transplants or other surgical procedures, and rarely if ever leading to secondary transmission), some examples are known in other species. These include the canine transmissible venereal tumor, a sexually transmitted cancer that infects dogs, coyotes and related species, and the devil facial tumor disease, a transmissible cancer that infects Tasmanian devils and is believed to be mainly spread by biting.

There are different reasons why cancer patients are mostly excluded from donating organs.

1. Although the probability of transmitting cancer is small, it is not zero. Also tumors tend to form metastases over time, which spread throughout the body. It is possible that a cancer has formed metastases in other, distant organs, which are too small to detect. Since recipients of organs have to take strong immune suppressing medication to prevent the rejection of the transplant, the immune system might not be able to fight the cancer cells anymore. Because of this medication, transplant patients already have a higher chance of getting cancer, because their immune system does not capture cells when they go rogue. See reference 1 for a more general description and 2 for the detailed report, reference 3 for the higher cancer risk of transplant patients.
2. Tumors can go dormant when they are treated with chemotherapeutics to evade destruction, but these dormant cells can re-wake. If this happens after the transplantation of an organ, this is obviously bad for the recipient. See reference 4 for details on mechanisms.
3. Cancer patients who died from the disease have often been treated with agressive drugs like chemotherapeutics. While these attack mostly the fast dividing cells of the tumor, they also have more or less strong side effects on the body. A lot of organs are simply not in the state (or have a strong chance of being in this state) anymore to be transplanted. Nobody wants to transplant a dysfunctional organ. See reference 5 for details.

References:

1. Cancer Spreads from Organ Donor to 4 People in 'Extraordinary' Case
2. Transmission of breast cancer by a single multiorgan donor to 4 transplant recipients
3. Cancer in the Transplant Recipient
4. Tumor dormancy as an alternative step in the development of chemoresistance and metastasis - clinical implications
5. Long-Term Side Effects of Cancer Treatment