At first, nuclear waste seems quite different from CO2. However, oddly enough they share that one thing that makes them quite the headache to deal with. Namely, long half lifes. CO2 that we pump out today will linger in the atmosphere for centuries, the CO2 half life is actually about 120 years (source). Why thats longer than most isotopes found in nuclear waste from a reactor, strontium-90 and cesium-27 are both around 30 years, while samarium-151 comes close at 94 years (source). These long half life’s make managing very tricky, what CO2 we pump out will hang around for a long time, so best burn as little stuff as possible.
The three main ways on how CO2 exits the atmosphere, most gets dissolved into the oceans (source), some is taken up by plant matter and winds up in a bog (source), and lastly we have very slow geological processes which work over thousands of years (source). After all there is a reason some call oil and gas “dino-juice”, as thats how long ago that carbon left the atmosphere.
Like radioactive waste, there are ways to manage the process. For example nuclear-waste can be re-processed and usable nuclear-fuel extracted (source). Carbon capture is also an option for CO2 sequestration, if stuff absolutely must be burned, there are ways to mitigate the impact, beyond of course, burning as little as possible.
Wetland restoration is one of the cheapest ways to accelerate carbon sequestration (source). For as little as $1, a tonne of CO2 is removed from the atmosphere via the magic of carbon sequestration.
Tree-planting is another common way to speed things along. Perhaps $5 to $10 would do to offset a tonne of CO2 (source). It can be particularly cost effective if there is say vacant farm-land which is no longer being used, or other marginal land that can be returned to a forrest, assuming of course that wetlands is not an option (source). Further, tricks like putting some of the dead trees away in a “wood-vault” might be clever ways to turbo charge the process (source).
While we do have ways to speed CO2 sequestration, they all carry some cost and inconvenience to them. Best to avoid burning stuff if you can, as Carbon removal processes are quite slow, besides oil in the ground is not a bad way to store carbon, so why burn it if you do not have to.