What if humanity burns all fossil fuels it can find, putting the negative effects of climate change aside, would the enormous amount of CO2 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
No. For a start, humanity will not burn all the fossil fuels it can find. But let's suppose for the sake of argument that humanity recognises no restrictions and fulfils all its energy needs by burning fossil fuels. Only a limited amount of fossil fuels are recoverable; the vast majority will have to stay in the ground.
Oil, for example, won't suddenly run out. What would happen is that it would gradually become scarcer, and increasingly expensive to recover. Eventually it would become too expensive to burn, so humans would have to turn to other fossil fuels, coal being the obvious one. Some would be used to make synthetic oil, and some burned in its natural state. The world has very large recoverable reserves of coal There could be enough to last humanity for a thousand years, but eventually recoverable reserves would become so scarce that it too would become too expensive to burn and, like oil, would be used as a source of chemicals and plastics.
All these fuels together, oil, gas and coal, might conceivably be burned for 2,000 years before becoming too scarce and expensive to be used as fuels. During that time,plants,especially in the oceans, would not be idle. Much of the CO2 generated would be sequestered by natural means, and although there would be a big rise in atmospheric CO2 with consequent climate warming, it wouldn't be nearly enough to make the atmosphere unbreathable.