How does the introduction of living things into a closed system affect the rate of change of entropy? Does the introduction of living things into a closed system increase or decrease the overall rate of change of entropy of a system?
 A: There is clearly no universal answer. One would have to say what living things, and in what state, have been introduced.
Life forms are generally capable of reducing their own entropy, but of course they increase the entropy of their environment by an even larger amount because no one, not even life forms, have the right to violate the second law of thermodynamics (saying that the total entropy never decreases).
So living things are able to cool them down, order them, and so on. However, we must realize that the numerical value of the entropy drop coming from organized or symmetric matter – e.g. from the symmetry of an egg – is very small. Moreover, there is nothing really mysterious or "vitalistic" about the effect of life on entropy. For example, the ability of the organisms to cool themselves down by sweating is shared by refrigerators.
If one really calculates the rate of change of the entropy, living things generally create heat by metabolism and this heat creation is the main term that influences the entropy, too. Once again, there is nothing terribly special or mysterious about heat arising from metabolism in living forms; fire is capable of increasing the entropy via produced heat, too.
A: Maybe we can consider a scenario. Suppose a human is introduced into the system. What if they have the ability to halt the largest entropic processes not directly related to their own survival? An example would be if there is fire in the closed system, the human might be able to stop the fire, thereby greatly reducing the entropy increase (proportional to the original magnitude of the fire). If the human's own metabolism incurs a smaller amount of entropic change than the fire was doing previously, then you have a case where this system's entropy has slowed its increase. 
A: It is possible that in some very very special case the overall rate decreases, but in general it will increase.
Consider a mature living system, its thermodynamic state is well-approximated by a stationary state, which means that entropy does not change $\Delta S=0$. Now using the DeDonder decomposition
$$\Delta S= \Delta_\mathrm{i} S + \Delta_\mathrm{e} S$$
and the second law
$$\Delta_\mathrm{i} S \geq 0$$
We obtain the well-known relation $\Delta_\mathrm{e} S \leq 0$ that proves that living systems are dissipative structures which need a negative flow of entropy to survive. I suppose that by "closed" you really mean isolated (i.e. neither matter nor energy exchange). If the living system is isolated then the flow of entropy is zero $\Delta_\mathrm{e} S = 0$, cannot compensate the entropy generated by dissipative processes in its interior, and dies
$$\Delta S= \Delta_\mathrm{i} S \geq 0$$
A: Intelligent living things -despite of there behavior as a machines that decreasing their own entropy by producing external work and increasing environment entropy as was described by 
Luboš Motl- can also decrease the entropy locally, see for example Boltzmann Brain Paradox and Maxwell Demon, anyway such statements still speculative to some extent and not resolved yet fully.
