Why can radiocarbon dating only be used for things that were once living organisms?

Short Answer:

Radiocarbon dating can only be used to date an object that had a known quantity of carbon 14 at one time and still retains enough carbon 14 to measure.

Carbon 14 dating can only be used on objects which were once living things (plant or animal) because nothing else has a known starting composition of carbon 14 and is less than 50,000 or so years old. In older objects the carbon 14 has decayed to such a low level that the detection becomes difficult.

More:

Living creatures constantly exchange carbon in their bodies with carbon from the atmosphere and so the isotopic concentration of carbon 14 in a plant or animal is the same as the concentration of the atmosphere. That stops, of course, when the creature dies. Since carbon 14 is radioactive, the fraction of carbon 14 in the remains of the creature will decrease over thousands of years. By measuring how much is left, the date the the creature died can be measured with some degree of accuracy.