Extracting CO2 from the air and turning it into a burnable fuel is dead easy. Plants do it all the time. Live plants are called "animal food", and dead plants are called "fossil fuel". So if you want less CO2 in the atmosphere, grow more trees and breed fewer humans and animals. Costs nothing, achieves everything, and no new technology required.
Just because plants can do it doesn't mean it's easy. There is also a problem of scale--there is no way for the plant life on this planet to keep up with the rate at which we are burning fossil fuels.
I refer to this chart of atmospheric CO2 concentration (http://www.mongabay.com/images/2006/graphs/co2_mauna_loa.jpg). All of the parts of the curve with negative slope are due to photosynthesis--there are more plants in the Northern hemisphere, so more carbon dioxide is absorbed April–September. I am slightly optimistic by how steeply sloped downward these portions of the curve are, but then the portions where the curve goes back up again are because of the decay of leaves and fruit etc. during the fall and winter, so unless we have some way of collecting all of the plant material and turning it into fuel, plants are not the way to go. Overall this cycle is roughly carbon neutral (ever so slightly carbon negative--absorbing a little more than is released, but only because of organic matter that gets subducted before it can decay--making fossil fuels), and the overarching trend of increasing CO2 is largely due to burning of fossil fuels.
There is a lot of carbon still in the ground (every molecule of oxygen in the atmosphere corresponds to an atom of carbon stored somewhere), but this took billions of years, and at the rate we're going (~200 ppm/century), it would only take 100,000 years to burn all of that, so we are releasing carbon almost 4 orders of magnitude faster than it is being sequestered.
Maybe some serious bio-engineering can help the plants catch up, but I doubt it.