To stop anthropogenic climate change, is it necessary to stop extracting petroleum and start pumping carbon under ground?
Summary
Most of the remaining stocks of hydrocarbons (coal, natural gas, oil) will have to remain unburnt. In almost all cases, that will mean leaving them in the ground. We already have proven technology to prevent new emissions of anthropogenic greenhouse gas emissions, and in most cases, the technical and economic barriers are solved: the remaining problem is purely political.
Let's make sure we're talking about the same thing
Let's get our terminology straight first. The big issue is rapid climate change; and we can do something about that, because the current rapid climate change we are starting to experience, is anthropogenic - it is human-caused.  So, &quot;climate change&quot; here refers not to historic natural processes over thousands to millions of years, but the present anthropocentric-induced processes that are happening over decades. That is the most common usage of the phrase these days, but it still can be useful to state it explicitly once in a while so that we know we are discussing the same thing.
Stocks and flows
Climate change is a result of changes in the stock of greenhouse gases in the atmosphere.
Those stocks have increased a lot over the last 180 years or so, and are continuing to change, because we've messed about with the flows. In particular, we've increased the rate of emissions from sources, without increasing the rate of take-up of sinks.
The bathtub analogy: we've opened the bath taps further, so water's flowing in faster. But we haven't widened the drain, so the level of water in the bath is rising, and soon will overflow and flood the house. It's not the rate at which water enters the bath that's the problem in and of itself: it's the level of water in the bath, and that's rising quickly because the inflow is much faster than the outflow. Similarly, what matters for climate and climate change is the level of greenhouse gases in the atmosphere, and the level of CO2 absorbed into the oceans.
Even if new net emissions went to zero this year (technically impossible without the collapse of human civilisation, which would rather defeat the whole exercise), we'd still see increasing heat content of the Earth over the next few years, as things move towards equilibrium, based on the current levels of greenhouse gases which represent a historic increase over the levels that prevailed over the last few thousand years during which human civilisation has developed.
So, to prevent further climate change, we have to balance out sources and sinks. So we can do either, or a combination, of two things: we can decrease the rate of emissions from sources down to the level where existing sinks can absorb them; and we can increase the rate at which sinks absorb.
Reducing sources
Stopping burning fossil fuels is the easiest and simplest way to decrease the rate of emissions. To the best of my knowledge, we now have alternatives for almost all of the major emissions sources, with the possible exception of methane emissions from livestock farming. Some are very well-established, like hydro and onshore wind electricity generation. Some are maturing quickly, like PV, offshore wind. Some are successful in the lab, and ready to move to commercial-scale prototypes, such as decarbonised production of steel, clinker, and liquid hydrocarbons.
Increasing sinks
Carbon sequestration is one way to increase the sinks. It's been trialled at small scale, and remains problematic. The way that's been done so far is to pump the carbon dioxide underground. At the moment, no business has been willing to accept the open-ended liability that is posed by the risk of leakage. So at the moment, the market is saying that this is not a viable technology. That might change. It's also worth noting that at the moment, the market also wants to use CO2 sequestration as a means of extracting even more oil (Enhanced Oil Recovery, EOR), which just makes the whole problem worse.
One way that is fast, effective and affordable is increasing the amount of carbon in biomass. Reforestation and afforestation are two ways to do this, and would help reverse a lot of the human-caused loss of forest ecosystems, if done well.