The carbon reaction goes like this: Carbon + Oxygen = CO2 + energy.
You can reverse it, like you can with any chemical reaction, like this: CO2 + energy = Carbon + Oxygen
This is all high school chemistry stuff. But the long and short is: To sequester carbon from the atmosphere requires at least as much energy as you got by burning the carbon in the first place.
That's why no scheme to unburn the carbon can ever work.
Also, "So far"? If you believe that we're ever going to break the conservation of mass and energy, you literally believe in magic.
You can burn carbon at one location, and unburn it elsewhere. For instance, you could burn it on planes, which benefit directly from fossil fuels' high energy density, and then unburn it in areas with surplus of solar or wind power, taking energy from renewable sources.
Here, no thermodynamics violated, and yet the system allows us to eat our cake (high-density energy storage) and eat it too (net zero carbon emissions).
Why? How does it follow from the previous sentence?
So what?
You're implicitly checking if energy is "lost", for some definition of energy loss. But that's not what's discussed here. Nobody's proposing a perpetuum mobile, we're just talking about environment. The question is, is it better to not burn in the first place or burn and then clean up? And the answer, perhaps surprisingly, is not obvious.