I suppose there is a sort of Anna Karenina principle for scientific error: every correct scientific argument is the same, but every erroneous argument is erroneous in its own way, but there are some general strategies: put in some correct stuff, put in some related equations but at the critical moment equate some things that aren't actually equal or derive something from an inappropriate equation. Above all, though, make your argument complicated, so that it's really hard to see exactly where you went wrong.
In this last respect I can't say that ex-Professor Motl's Post is a model, because he's really quite clear in his argument and it's pretty obvious exactly where he goes off the tracks in his calculation of climate sensitivity. Of course he does emit a lot of smoke after that point, but it's way to late too hide the dirty deed by then.
To set the scene, let's remember that CO2 causes planetary warming by increasing the outgoing impedance to radiative heat flow through the atmosphere without much affecting the incoming radiation. Since the amount of solar radiation reaching the Earth's surface is approximately unchanged, the amount of radiation leaving the top of the troposphere has to remain unchanged as well, but the increase in outgoing impedance means that the surface warm up until the temperature gradient is large enough to drive an outgoing heat flow that matches the incoming.
The climate sensitivity parameter attempts quantify the surface temperature change needed to compensate for the increased outgoing impedance in terms of the so called radiative forcing:
The radiative forcing of the surface-troposphere system due to the perturbation in or the introduction of an agent (say, a change in greenhouse gas concentrations) is the change in net (down minus up) irradiance (solar plus long-wave; in Wm-2) at the tropopause AFTER allowing for stratospheric temperatures to readjust to radiative equilibrium, but with surface and tropospheric temperatures and state held fixed at the unperturbed values.
Lumo next writes down the Stefan-Boltzman law for the dependence of total radiation on the temperature but makes his first critical error in equating that to the radiative forcing - or at least using the same symbol (RF) for it. This fundamental confusion becomes the source of all his subsequent errors. If all the radiation emitted by the surface reached the top of the atmosphere, this formulation would be correct (and there would be no greenhouse effect), but it doesn't, as Lubos explicitly admits. CO2 and other components of the atmosphere ensure that only a fraction of the outgoing radiation (or more generally, heat - since a lot of the transport is not radiative) reaches the top of the atmosphere. The change in that fraction due to addition of a component (such as CO2) is the radiative forcing.
Despite his first fundamental error, Lumo acknowledges the CO2 impedance and calculates (or at least acknowledges some calculation to get) a radiative forcing due to the CO2, getting approximately the IPCC result. What's been left out at this point are the feedbacks. If CO2 warms the surface of the Earth, that warming will increase the amount of water vapor and methane in the atmosphere. They too will reduce the fraction of outgoing emissions that reach the top of the atmosphere.
Here though, Lumo makes his second and more bizarre error. He appears to assume that the effect of the feedback is to change the shape of the curve of dependence of surface emission upon temperature. Of course it doesn't have any such effect - what it does do is change the impedance of the lower atmosphere to outward radiative heat flow. The S-B law governs emission - not net outflow, and the emission must increase (via increasing surface temperature) to compensate for the decrease in atmospheric transmittance.
Next he wanders off into some deductions from his peculiar error - I'm afraid that I didn't try to follow them.