If stars were like cars, then they would burn their core hydrogen until they ran out and the star would fade out. But fusion converts hydrogen into helium. As the helium ash builds up, energy generation stops in the core. The fusion process moves outward into a shell surrounding the hot helium core. For small stars, this temperature is never reached and the helium core remains inert.
Fusion alters the chemical composition of stellar interiors.
Note that since the cores of stars are so large and massive, it takes anywhere from 100's of thousands to billions of years to run out of hydrogen fuel. Clearly, stars that burn brightest, burn fastest and, thus, have the shortest lifetimes. From thermal equilibrium we know that the temperature will then begin to drop and then the pressure will also decrease in the fusion core. From hydrostatic equilibrium, we know that a drop in pressure means that the core region of the star will contract slightly. At this point, the hydrogen burning shell becomes important as the sole source of energy in the dying star.