Need to look into this a bit more, but what would you say the theoretical efficiency is, could we reach a point where you can actually burn fossil fuel to net extract CO2 via direct air capture or another sequestration method that can be scaled?
I don't know enough about thermodynamics to calculate the fundamental limits. I suspect that low-temperature sorbents like triethanolamine can currently do direct air capture for less than the energy produced by the fuel, but the process is complicated, involving things like embodied energy in fan motors and hard-to-predict maintenance costs. In a cousin comment (https://news.ycombinator.com/item?id=44461370) I did an upper-bound calculation with a very-well-understood atmospheric carbon capture process that people have been doing inadvertently in a no-net-sequestration fashion for thousands of years. It came up with burning no more than 2 kg of coal per kg of carbon dioxide removed, which is theoretically significant net sequestration (if you do point-source capture on the coal burning) but far too expensive to make a dent.
Sequestration processes like serpentinization are actually exothermic. The idea there is that you react carbon dioxide with olivine and get serpentine and heat. There's vastly more olivine available than any crustal rock such as limestone. You can do this in lots of ways; for example, you can pump concentrated carbon dioxide down a well into fracked olivine, where it reacts, or you can crush olivine from an olivine quarry into olivine sand and just dump it on beaches, or build artificial islands out of it in Dubai or disputed areas of the South China Sea. Crushing the olivine costs energy, though, and it's energy that's mostly not stored for the process; it just provides more surface area for the same reaction.
I don't actually favor this approach (what do you do if you decide the beaches are removing too much carbon dioxide? Beaches don't have emergency stop buttons) but it does show that in principle the effective energy consumption of direct air capture doesn't even have to be positive.