I suppose that chainsaws contribute to the destabilization of the atmosphere because they allow people to quickly and easily cut down trees. While all plants help to regulate the levels of carbon and oxygen in the atmosphere, trees are the largest of plants and contribute to this process more than any other terrestrial plant.
Chainsaws, and logging, in general, has led to the destruction of forests across the globe. Forests are a huge carbon sink. Trees and other plants absorb carbon from the atmosphere and using the process of photosynthesis converts it into plant material. When trees are cut down, less of this carbon can be absorbed from the atmosphere and stored inside these plants. Also, once forests have been removed from the landscape, the exposed ground absorbs more heat than the leaf cover would, further leading to temperature rise.
There also is some greenhouse emission output from chainsaws themselves. While electric chainsaws do exist, most are gas-powered. Emission standards on smaller appliances are not as strict as they are on large vehicles and the emissions produced by gas-powered chainsaws do add to the overall amount of carbon in the atmosphere. Furthermore, the emissions from chainsaws tend to put out relatively large particulate matter into the air.