Why does radiocarbon dating only work in nonliving creatures?
There are plenty of good sources online explaining the principle behind radiocarbon dating. For instance, the wikipedia explains:


  During its life, a plant or animal is in equilibrium with its surroundings by exchanging carbon either with the atmosphere, or through its diet. It will therefore have the same proportion of $\ce{^14C}$ as the atmosphere, or in the case of marine animals or plants, with the ocean. Once it dies, it ceases to acquire $\ce{^14C}$, but the $\ce{^14C}$ within its biological material at that time will continue to decay, and so the ratio of $\ce{^14C}$ to $\ce{^12C}$ in its remains will gradually decrease. Because $\ce{^14C}$ decays at a known rate, the proportion of radiocarbon can be used to determine how long it has been since a given sample stopped exchanging carbon – the older the sample, the less $\ce{^14C}$ will be left. [16]


Your question is somewhat subtle: why doesn't decay begin immediately once the isotopes have been incorporated into an organism? Can't measurements of part of an organism tell us when that particular organ or body part was formed? 

The key limitation is explained in a document from a company (Beta Analytic) that provides biobased content / renewable carbon measurements:


  Although carbon-14 is radioactively decaying away in the body, it is constantly being replaced by new photosynthesis or the ingestion of food, leaving the amount relatively constant. 


But if a body part no longer cycles carbon (is in effect dead), then radiocarbon dating should be useful. One application, for instance, is in dendrochronology, the analysis of tree growth rings to assess age and location:


  Dendrochronology is useful for determining the precise age of samples, especially those that are too recent for radiocarbon dating, which always produces a range rather than an exact date, to be very accurate. However, for a precise date of the death of the tree a full sample to the edge is needed, which most trimmed timber will not provide. It also gives data on the timing of events and rates of change in the environment (most prominently climate) and also in wood found in archaeology or works of art and architecture, such as old panel paintings. It is also used as a check in radiocarbon dating to calibrate radiocarbon ages.1


Combining tree samples of known age (thanks to dendrochronology) and other sources with radiocarbon ratios provides a calibration method.