Yes. The Sun can run out of expendable energy.
~ It will run out of hydrogen that can be converted to helium in about five billion years.
~ After that it will be a few more billion years before other elements that can fuse and provide energy are exhausted.
~ It would then take the Sun trillions of years to cool to where no light is emitted. This may not occur because in about 22 billion years from now, according to some theories, the expansion of the universe may tear it apart in an end call the Big Rip.

Even after the Sun stops emitting any light, it will still exist. If there is no Big Rip, the subsequent time of existence of our Sun is thought to be on the order of the time of existence of the proton, known to be at least 10^32 (ten with 32 zeros after it.) years. Some simple grand unified theories predict a proton lifetime of no more than 10^49 years. If these theories are not valid, the proton may decay by more complicated nuclear processes, or by quantum gravitational processes involving a virtual black hole; in these cases, the time of existence is estimated to be no more than 10^200 years. If protons do decay, the mass of our Sun will decrease very slowly with time as its nuclei decay, until it loses enough mass to become a non-degenerate lump of matter, and finally disappears completely.
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