Why are cells affected by time?
Individual cells don't live forever because they are damaged by their environment, for example by chemicals, radiation and mechanical wear and tear, or by other organisms (see this review). In multicellular organisms, such "worn out" cells are typically dismantled by programmed cell death and new cells are generated to take their place.

The life span of animal cells varies widely depending on their location and function. For example, epithelial cells lining the skin and intestines are exposed to a harsh environment, and consequently live only a couple of days. At the other extreme, neurons, which are safely lodged in the protected brain tissue, can live as long as the individual (or even longer). 

Note that the life span of a given cell is very different from replicative age, which refers to the number of divisions a cell lineage has undergone, measured from some progenitor cell. Most animal cells are allowed only a limited number of cell divisions (known as the Hayflick limit) and thus their lineage is "mortal", while that of microorganisms are not.