It's getting a bit late, I can reply tomorrow in greater detail if you want, unless someone else comes up with a detailed answer before that. But the the general gist is that the carbon cycle of plants is.. well.. a cycle. Most of the CO2 that is removed from the atmosphere by photosynthesis of plants is eventually returned to the atmosphere. As plants die the organic matter they produced will decompose - aerobic decomposing/digestion produces CO2 and anaerobic produces mainly methane. And methane eventually reacts with oxygen to form CO2 and water. Some amount of carbon gets deposited in the soil permanently (eventually forming things like coal and oil), but that's a relatively minor part of the carbon cycle on land. However by increasing the total amount of living plant biomass you do decrease the amount of CO2 in the atmosphere. Then all we gotta do is take all those plants and shove them deep underground so that the carbon is removed from the active cycle.. reverse mining, heh.
That's a good point I hadn't really thought about plants decaying and releasing it back out, and really never think about methane. I guess the reverse mining would give us a reason to use all those old abandoned and boarded up mines in the South Eastern US.
It gives a nice contrast thinking the world is currently consuming 20 million metric tons of coal (source) and 0.0139 km^3 of crude oil every day (87 356 000 barrels per day, source). That's 5500 olympic sized swimming pools of crude oil every day.. I'm not sure what is the mean density of coal but guessing 800 kg/m^3 that's 25 million cubic meters, which in turn is roughly equal to 10000 olympic swimming pools or 10 Great Pyramid of Gizas Every day. With an increasing pace.
I'm not so great with math stuff, it was never my strongest subject despite it's real world need so the Olympic swimming pools and pyramid references really helped me with this. We are definitely in need of some changes to our thinking these days based on the responses here.