Cross species transmission of cancers would be very low, and as Supercryptid mentioned, the tissue would have to be very very fresh (live).
Biologists do infect rats with Human cancers (through surgery). However, it is only possible to do so with seriously immune compromised rats called "Nude Rats". Otherwise the natural immune response would be to fight the foreign tissue.
Potentially AIDS patients, transplant patients, or other patients with serious immune deficiencies would be at higher risk.
This brings up a couple of points. If a person has cancer, or a history of cancer, then they would be automatically excluded from being an organ donor. Since cancer can spread in the blood, then even unaffected organs would be excluded from transplant. Likewise, it would be bad for a cancer patient to donate blood.
HPV is a highly communicable viral disease that that carries the risk of developing cervical cancer. While men can be carriers of the virus, the cancer risk is believed only to be in women. However, the transmission is likely just creating an environment where cancer can develop, without transmitting the cancer itself.
HPV has been recently linked to some kinds of cancer in men, especially oral cancers.
Oral cancers have seen a sharp rise (in both men and women) in the last 30 years, especially in young people. It's thought this might be because oral sex is considered safe sex (as far as AIDS goes).