Why do human cells have a "death" date while cancer cells seem to live perfectly fine without that limitation?

Immortal cells survive well, but they do not function. A liver cell functions to remove impurities from your bloodstream. A cancerous liver cell only multiplies. If a normal cell starts to lose its ability to filter your blood, your liver function decreases, so it is advantageous for the cells to die, be carried away, and a new cell to take its place. Since cancer cells really only multiply, crowding out normal cells, it does not matter if they get less efficient at multiplying, since new cancer cells are being produced all the time, growing the overall mass.

For a given tissue or organ to function correctly cell population needs to be controlled. An organ needs to maintain a constant size while replenishing cells as they die. Cancer cells are very hardy, but people with cancer are not so fortunate. Cancer cells often reduce the functionality of the organ or tissue where they originate. They grow unchecked. This leads to physical problems. Cancer cells become less specialized. They spend energy dividing instead of spending it performing their proper function. Sometimes they do very abnormal things. Perhaps the most dangerous property of cancer cells is that they can metastasize. They can move through the circulatory system to other locations in the body.

If cells didn't die we would just keep growing and be like tumours

cancer cells arent fine they are out of control!

Cancer cells die when their host dies.