I'm assuming we can neglect differences due to radioactivity/radioisotopes in biological systems.
Hydrogen vs Deuterium
The kinetic isotope effect - the difference in reaction rates and reactivity for different isotopes of the same element - is well known for hydrogen and its two stable isotopes protium and deuterium. For example levels of deuteration above 25% have shown adverse effects in a variety of animals. [1]
Heavy Elements
But stable isotopes of elements as heavy as carbon can have measurably different effects in biological systems.
In photosynthesis, lighter $\ce{^12C}$ is incorporated into plants faster than $\ce{^13C}$ creating a difference in the abundance of $\ce{^13C}$ in biological sources compared to other enviromental sources. Furthermore different methods of photosynthesis - i.e. the $\ce{C_3}$ and $\ce{C_4}$ pathways found in different groups of plants - lead to different discrepencies in $\ce{^13C}$ abundance ($\delta^{13}\ce{C}$) which can provide information about the diets and nutrient sources of the animals that eat them.