Once an organism dies the carbon is no longer replaced.
This fact should always be remembered when using radiocarbon dates.
At an archaeological dig, a piece of wooden tool is unearthed and the archaeologist finds it to be 5,000 years old.
A child mummy is found high in the Andes and the archaeologist says the child lived more than 2,000 years ago.
Obviously there will usually be a loss of stable carbon too but the proportion of radiocarbon to stable carbon will reduce according to the exponential decay law: R = A exp(-T/8033) where R is C ratio of the living organism and T is the amount of time that has passed since the death of the organism.