How does combustion of the biosphere affect the atmosphere?
I'll break down the question: combustion refers to burning, which, as a base reaction releases CO2 and H20, but in reality, combustion reactions release much more. The biosphere refers to all life on earth, or all areas colonized by life forms. So, put more simply, your question asks what burning living or formerly alive things does to the atmosphere.
Coal, oil, and natural gas are all fossil fuels, which are essentially compressed and decayed organic matter. When we burn them they release chemicals like CO2, SO2, N2O, etc.. While SO2 is not a GHG, merely an air pollutant, CO2 and N2O are major GHG's that cause the majority of anthropogenic climate change. Since all life is carbon-based, burning most members of the biosphere will release CO2 and will thus contribute to climate change.
This is probably the most major effect. Other impacts on the atmosphere are the release of VOC's (volatile organic compounds) that are released when burning things like trees, and the release of particulate matter when burning most organisms, which can impact air quality for all surrounding organisms. These two effects are only relevant in the atmosphere close to the ground while the climate change impact affects higher up in the atmosphere.
Summary: climate change, release of gases and chemicals into the atmosphere such as VOC's and particulates.