What would happen if the carbon cycle stopped?

Organic compounds like carbohydrates, lipids, proteins and nucleic acids all contain the element carbon. Carbon must cycle between the abiotic (non-living) and biotic (living) components within an ecosystem or that ecosystem would cease to function.
Consider carbon dioxide (CO2) in the atmosphere. Plants and algae take in carbon dioxide, along with sunlight and water and carry out photosynthesis. The carbon from the carbon dioxide is incorporated into a molecule known as glucose--(C6H12O6) which is produced by autotrophs like green plants.
Once glucose is produced in plant leaves, consumers can eat the plant to obtain energy. Meat-eaters can eat the plant-eater all the while transferring carbon from one consumer to the next in a food chain. However, through the processes of animal and plant respiration, carbon dioxide is released once again to the atmosphere as a waste product. 
As organisms die and decompose, carbon dioxide is produced and returns to the environment. Ancient partly decomposed plants that were buried under water became coal, and marine plankton remains that were buried became natural gas and oil over a long period of time. Coal, oil and gas are fossil fuels that contain carbon because they were derived from living organisms whose bodies contained carbon compounds. When combustion occurs, carbon dioxide is released into the atmosphere.
Some carbon dioxide is absorbed into the oceans and after the processes of burial and compaction, it becomes a component of limestone which is calcium carbonate and may become incorporated into coral reefs.
To summarize- the carbon cycle is vital because carbon is present in important organic compounds found in living organisms. It must be able to cycle between the living and non-living portions of the ecosystem. Without adequate carbon the ecosystem would be unable to function.