Human cells do eat human cells. In fact the process of cell-eating is called phagocytosis, and the eater is known as the phagocyte. Phagocytes are commonly referred to as white blood cells.
Dead cells can come in the form of many things. I classify them into 2 groups, externally and internally.
Firstly, skin shedding is the process in which our dead skin cells get away from our body. They are removed by bathing or scratching. Sometimes, they just drop off or get blown away by the wind. That's why, wherever you go, you are actually leaving bits and pieces of yourself everywhere. The body continuously gets rid of dead cells and scientists estimate that by 7 years, your body would have replaced all old cells into new ones. It can be said that you are a new you. Unfortunately, our brain cells do not die away and get replaced.....pity we don't get smarter.
Internal dead cells refer to the digestive track, in which the cells lining our alimentary canal drop of, and get carried away by the poop. Our poop mainly consists of dead cells that have attached themselves to it, undigested food (fiber) and bacteria. You see, pooping does not relieve you of a tummy ache. It helps cleanse your body.
Next, not all cells which are dead get disposed off. Some living cells, such as red blood cells gets broken down in the spleen. When they are broken down, the materials that made it can still be reused. For example, when RBCs are broken down, hemoglobin is released and further broken down into iron which is stored in the liver. Phagocytes play a part in the breaking down of such substances.