The assumption that CO2 levels will continue to rise if we keep burning fossil fuels requires that the natural systems that comprise the Carbon cycle are effectively static and cannot respond to changes dynamically. This just isn't the case, we can see this already with CO2 fertilization causing significant global greening. Plants are already responding to increasing CO2 levels. Oceans will also respond, along with the life that lives within them, phytoplankton is a significant consumer of CO2. Earth is currently very CO2 starved, increasing CO2 levels will result in an explosion in the abundance and productivity of life.
The most important aspect though will be Ocean temperatures. The Oceans and Sun drive medium term climate on Earth, with longer term changes driven by orbital mechanics within the Solar System and even longer term changes related to galactic scale orbital mechanics (e.g. the transit of the Solar System through the galactic arm significantly increases cosmic rays, causing greater cloud formation, etc). But, Ocean temperatures will determine what happens with CO2. Through Henry's Law, the solubility of gasses, including CO2, is proportional to the temperature of the water. Cooler water can absorb more CO2 than warmer water. This is why the ice core temperature and CO2 proxy data shows a lag of CO2 to temperature changes of ~700 years. Temperature change drives CO2 levels, not the other way around. The various cycles that drive the climate are natural and there's nothing we can do to change them.
Even if CO2 levels were to rise to 2000ppm or even higher, this would be no problem for animal life. The cognitive effects of increased CO2 levels are largely due to acute changes in CO2 level. Gradual changes can easily be adapted to. You can happily survive CO2 levels over 5000ppm. Any large group of people in an enclosed space will result in highly elevated CO2 levels, often exceeding 5000ppm.
CO2 emissions don't change the temperature. The activities of Man can change the climate though, but mainly through land use change. Clearing forests to create farm land, building cities, damming rivers to create lakes were none previously existed, irrigating vast areas, etc. All these activities create significant local or regional climate change, and together when conducted across large portions of the land of Earth, can create global climate change. Though the resulting effects will be very difficult to predict as the climate is a complex system. This cannot be predicted even 10 years into the future, attempting to predict what the Earth's climate will be in 3000 years is impossible, errors in the input data propagate forward and accumulate, so you might be able to say the temperature will be 17C +/- 5000C in 3000 years (which is obviously meaningless).
TL;DR; Climate is a complex system, it is driven by natural cycles and these cycles will (and are) react to changes in CO2 level from any cause. Water is the key to Earth's climate, in all it's forms it controls the climate. The properties of water are so awesome that it's close to miraculous.