The rates at which cells are replaced vary quite a bit. For example, in the inner lining of the small intestine, cells turn over in a week or less. In the pancreas, the turnover time may be as long as a year or more.
Sometimes the rate of cell division to replace cells depends on the state or size of the tissue. Liver cells rarely divide, but if there is injury to the liver and the liver is somehow reduced, liver cells will divide to get the liver back to the right size. Similarly, in the skin, the basal cells will divide only to keep the skin at a certain thickness. If the outer layers of dead skin peel away more frequently, the basal cells on the inside will divide more frequently to make up for the lost skin. If a person takes good care of his or her skin and there is little damage and fewer cells to replace, the basal cells will reduce the frequency of division.
For more information on tissue maintenance, I recommend the following textbook: