I suppose you mean that diamonds, being relatively indestructible by ordinary means, will not be 'recycled' into other forms of carbon?
I suppose further that the basic answer is 'no.' You can burn diamonds if you have a hot enough flame--1520 F is mentioned on one website. But under most probable scenarios, they will be around essentially forever--they won't decay or degrade, and if broken or crushed you just get smaller diamonds. So diamonds aren't being 'recycled' in any meaningful way.
But diamonds won't 'take over' the carbon cycle for 2 reasons:
1) they form very slowly--one estimate says diamonds in general may be 1/4 of the age of the Earth itself. And they form deep in the Earth's mantle, so their carbon was 'already withdrawn' from the carbon cycle.
2) The amount of carbon in diamonds is a tiny fraction of the total carbon in the world. The total diamond production of South Africa, the world's leading producer, is about 10 million carats. In kilograms, that's about 5,000, if I haven't dropped a decimal somewhere. (A carat is .2 grams.) By contrast, world emissions of carbon due to fossil fuel combustion reached 10 billion tonnes in 2010. In kilograms, that would be 10 followed by 12 'zeros.' So the ratio of diamonds to carbon emissions would be about 2 billion to 1--that's roughly you (or I) compared to the total populations of India, the US, Indonesia and Brazil. And human emissions are much smaller than the total carbon in the biosphere, let alone all the geologically-stored carbon.
So more carbon is being (re-)introduced into the cycle all the time by human activities. There's no danger of running out--though there is a clear danger that we will continued to disrupt our climate with all the CO2, which is a greenhouse gas.