This is a great question - maybe you will be a doctor or a biomedical researcher one day! Let's start out with a basic conversation, then for those with interest, a bit more detail.

Sometimes, a disease can be pretty scary. I bet you have had a cold before, right? That is not too scary, just uncomfortable. A cold is caused by a virus, and when you sneeze or cough, that virus can be passed to another person, and then that person may also get a cold. Influenza (flu) is the same. The same with bacteria that cause disease. These are called infectious diseases because they can pass from person to person and are caused by microbes.

Cancer is a very different kind of disease. Our bodies are made of cells and throughout our lifetimes, these cells must be renewed - they divide, but in a very controlled manner. If a mistake in this division process is detected, these cells usually die off or our immune systems can take care of it. Sometimes though, the "mistake" is undetected and the cells can divide in a rapid, uncontrolled manner - this is called "cancer."

The cancerous cells are normal cells that are out of control, and they still rely on the body for oxygen and nutrients. As the cancerous cells grow they can causes the organ or tissue they are located in to stop working properly and then, they can also spread to other parts of the body and do more damage in those places (this is called "metastasis").

So, to answer your question, the cancer cells die when the person dies (because those cells are part of the person's own body).

MORE INFO:
Because the "mistake" (mutations) that occurs in a cancer varies widely from person to person, it can be a challenge to treat the cancer. In the past, the treatments basically were poisons that would kill any rapidly dividing cell (including normal ones), leading to side effects. It's important to remember that "cancer" is a collection of diseases - a breast tumor is very different from a brain tumor, and must be treated differently. However, a breast cancer in one woman may also be quite different than one in another woman. Now, there is technology that allows a physician to determine the mutations associated with the cancer and there are some medicines that are targeted specifically to those mutations. This is called "precision" or "personalized" medicine. Some cancers, which were fatal just a decade ago, are now considered "maintenance diseases" because the therapies allow for long term survival with minimal side effects. An example of this is some forms of chronic myelogenous leukemia (CML).

Because it seems that biology has an exception to every generalization, including the one made above about infection .... it is true that there are some cancers that are associated with viral infections - human papilloma virus can cause cervical cancer in some women and this is why HPV vaccines are now recommended (they are very effective, by the way). We don't understand exactly why only a fraction of women exposed to the virus end up developing the cancer - a very high percentage of women in the US have been exposed to HPV.

In some animals, there are communicable cancers - a common one you may have heard of is feline leukemia virus (FLV). There is also an oral cancer that develops in Tasmanian Devils and it is passed from animal to animal (and is causing a crash of wild populations of these amazing creatures).

Interesting question. Luckily cancer is not contagious; cancer cells in one person cannot be transfers to another person. After death, the heart stops pumping oxygenated blood through the body. Normal cells and cancerous cells depend on oxygen to survive. The cancer cells die once they stop receiving oxygen. One possible way to treat cancer is to cut off the blood supply to cancer cells specifically.

Cancer is not a germ or a parasite like a tapeworm. Cancer is the body’s own cells that have gone bad. The cancer cells stop doing their job. They just keep dividing and making more bad cells. The bad cells crowd out the good cells that are trying to do their jobs.

When a person dies, all of their cells die, including the cancer cells. These cells can’t hurt anyone else.

You don’t catch cancer like you catch a cold. A person with lung cancer cannot give you lung cancer, whether they are alive or dead.

You probably know that when people handle dead bodies they protect themselves from germs and parasites by wearing gloves and other gear. They do not need these to protect themselves from cancer. They still wear them because anyone could have germs, no matter what they die from.

If you are interested in questions like this, you may want to study biology.

When a person dies, the cancer stays in the body and dies along with the person. Cancer is not an infectious disease like a cold or the flu and cannot transfer from one person to another. The cancer cells are human cells that try to make more copies of themselves even though it hurts the person. They are more like messed up human cells than a separate living thing that is trying to reproduce.

Cancer is a type of disease where cells behave differently than normal. Cancer cells stop doing what they are supposed to do and instead grow faster than normal cells. This makes it harder for normal body tissues to work – not only have cancer cells stopped doing their job but they take more resources and space, putting stress on healthy cells. Cancer can form in different parts of the body, and sometimes spreads from one type of tissue to another.

Even if they aren’t normal, cancer cells are still human cells and need the body to provide resources for them to survive. When the body dies, the cancer cells die too. Because cancer cells are part of a person, cancer does not pass from person to person like diseases caused by bacteria, viruses or other parasites. This is part of why it is so difficult to find and cure cancer – cancer cells can look like a person’s normal cells in many tests and looks/acts differently in each person. Many treatments (like radiation or chemotherapy) that kill cancer cells also kill normal cells too.

Scientists are still studying what causes cancer because the disease is very complex. They agree that cancer starts when cells make mistakes in their normal function. All cells make these mistakes and make them often. Healthy bodies recognize and correct them, but cancer forms when these mistakes are not corrected quickly enough. Some chemicals (like in cigarettes) and behaviors (like sunbathing) are said to “cause cancer” because they cause more mistakes to happen in cells or make it harder to correct those mistakes. Some people have genes that make it more or less likely for them to develop cancer, so what causes a particular person to develop cancer is difficult to know. Thankfully, scientists and doctors are making many advances in finding, understanding and treating cancer so that people can survive for much longer after cancer develops. Their ultimate goal is to prevent cancer from forming at all.

A particular cancer mutation actually is helping us understand and treat the disease. Because cancer cells need the body to survive, it is very difficult to test cancer treatments without risking hurting patients. Once removed from the patient, the cancer tissue dies, just like normal tissue. However, in 1951, Henrietta Lacks developed a form of ovarian cancer that was abnormal in a particular way that let her cancer cells grow in the laboratory. This was the first time that any human cells survived without needing a body! Her cells (called HeLa) have allowed scientists to test many treatments and drugs without hurting patients. They were used to develop vaccines for diseases like polio in 1952, and are still used to study cancer, AIDS, diabetes, etc. Henrietta Lacks’ cancer cells live on after her death but they don’t invade other bodies – they are helping us fight cancer and other diseases, making many people’s lives better.