It's worth noting that the Earth's atmosphere would be incompatible with current life at various past levels of atmospheric carbon.
In the past, Earth's atmosphere was almost entirely nitrogen and carbon dioxide[1], but at some point some plucky cyanobacteria learned how to photosynthesize and convert CO2 into O2, hugely changing the contents of the atmosphere [2] and killing a large amount of life that couldn't tolerate the new order.
So if we were to burn all the fossil fuels, it's quite possible that this would cause a climate catastrophe that would kill a large majority of life on earth. But there's a decent chance that the cyanobacteria and friends would survive and, after hundreds of millions of years, restart the evolutionary process.
Note that climate catastrophes take many forms, and the "greenhouse effect" does not even necessarily have to be involved. Take ocean acidification for instance. Atmospheric carbon raises the pH level of the ocean, which can cause cascading ecological collapse.
[1] https://en.wikipedia.org/wiki/Paleoclimatology
I think it's worth noting that on this timescale, we don't have forever:
"In about one billion years, the solar luminosity will be 10% higher than at present. This will cause the atmosphere to become a "moist greenhouse", resulting in a runaway evaporation of the oceans. As a likely consequence, plate tectonics will come to an end, and with them the entire carbon cycle... Four billion years from now, the increase in the Earth's surface temperature will cause a runaway greenhouse effect, heating the surface enough to melt it. By that point, all life on the Earth will be extinct." - from https://en.wikipedia.org/wiki/Future_of_Earth
I submit that this is probably Earth's only chance at a spacefaring civilization, and therefore Earth life's only chance at escaping the planet (and thus surviving the death of the planet). If we flub it and cause a mass extinction without anything to show for it, it is the beginning of the end of Earth's story.
http://burro.case.edu/Academics/USNA229/impactfromthedeep.pd...
There doesn't seem consensus on whether Earth could truly reach a runaway greenhouse state or would rather just settle at a higher temperature. In any case, only the ecological (and consequently economic) crises and refugees problem are the ones currently living people will be affected by, but that ought to be enough to take action.
[1] https://sciencing.com/effect-temperature-rate-photosynthesis...
In addition to the greenhouse effect, the location of the continents can also have a major influence on the climate due to the influence on ocean and air currents, the albedo, weathering speed of exposed rock in the tropics and different plant cover at different latitudes etc, thus it is hard draw good comparisons (only modelling is possible, and verification is hard).
To be fair, I do not believe we are capable of life extinction on Earth. We are also not able to make the conditions on this planet worse than it is on any other planet we can reach so far. But we can cause a mass extinction easily and make life for ourselves much much harder. Our situation is so optimized for the current conditions that changes will cause great pain. (Just look at the locations of cities, or how agriculture is run.) If we developed with the climate of 500 million years ago we would be just fine with that. The problem is the fast and large change.
[0] https://en.wikipedia.org/wiki/Carbon_dioxide_in_Earth%27s_at...
PS: One more point: CO2 causes some cognitive issues in humans in concentrations above 1000 ppm, which often occurs in meeting rooms and means the windows need to be opened regularly. It's not bad, you just get a little tired and dumber... so you take a break and open the windows. Now imagine the CO2 concentration outside is 5000 ppm (currently at ~415 ppm). We did not evolve for this.