Could burning all fossil fuels (available on earth) render the atmosphere unbreathable? What if humanity burns all fossil fuels it can find? Putting the negative effects of climate change aside, would the enormous amount of $\small\mathsf{CO_2}$ in the atmosphere eventually make it unbreathable?
Not directly.  What possibly could happen is that rising temperatures turn the parts of the ocean anoxic, which leads to large releases of hydrogen sulfide, which is toxic: https://www.chemistryworld.com/news/ocean-chemistry-changes-triggered-earths-greatest-extinction-event/2500368.article