You said they're removing it, then turning it into fuel. Which means it's going right back into the atmosphere once someone burns the fuel--still better than burning fossil fuels, by a long ways (basically, it puts cars on a carbon cycle like plants and animals are), but it's not going to substantially reduce the total amount of CO2 in the air. Instead, it's just going to slow (or, once enough of that is going on, stop) the *increase* in atmospheric CO2.
The odds that they would 1. remove enough to actually reduce atmospheric CO2 to an appreciable degree, and 2. not then burn at least most of it and re-release said CO2 are... rpetty small.
So, if you're asking as a practical question, don't worry.
If you're asking as a purely theoretical question... then let's look at what would happen if we had a magic CO2 vacuum that just sucked up CO2 and... removed it. Or at least spit back out oxygen, and just magically removed the carbon. Likely, plants would start to die off (which would increase atmospheric CO2 for a while). Animals would also be in trouble once their food started to die, especially the herbivores, but the scavengers, at least, would be good for quite a while longer.
It would also get a lot colder. Like, ice age cold, maybe even all the way to Snowball Earth levels (ie ice caps that reached or almost reached the equator). This would speed plant death, keeping the CO2 levels up for a little while longer, which would in turn slow the cooling a bit.
If humans planned for this, and stockpiled enough food and such while there was still enough plant and animal life, we would likely be fine until the concentrations got low enough that we started forgetting to breathe, which would likely (if we stored enough food for the occasion) be after all the plants were dead and every other animal, even the scavengers, ran out of food (assuming the magic CO2 vacuum was only removing, say, 1% of the total atmospheric CO2 every year). At that point, the Earth would become a lifeless ball of ice.