Assuming 2.3% per year exponential growth of human energy consumption (so, roughly speaking, that corresponds to multiplying consumption tenfold every century), it's argued that human annual energy consumption would be equal to annual solar energy output in 1400 years time. Is that correct?
For reference, human energy consumption is currently about $2 \times 10^{13} W$
(source: http://physics.ucsd.edu/do-the-math/2011/07/galactic-scale-energy/ )
I'll answer the current version of the answer and a bit of a previous.
The math is extremely basic. If the power production grows 2.3% a year and the initial power production is $2 \cdot 10^{13} \, \text W$ then in 1400 years your energy production would be:
$$P_{1400} = 1.023^{1400} \cdot 2 \cdot 10^{13} \approx 0.13 \cdot 10^{28} \, \text{W}$$
Now, Wikipedia states that Sun's energy production is 380 yottawatts ($3.8 \cdot 10^{26} W$). So as you see under the assumptions we will beat the sun by a factor of 3.4. I've glanced through the blog, the discrepancy is likely caused by the fact that the author actually started from $10^{12} \text W$ --- the energy consumption of US when the US's 2.3% trend was on.
So why would the Earth evaporate under the assumptions? Earth looses its heat by means of thermal radiation from its surface, just like the Sun. And if the Sun has its surface temperature around 6000 K to dissipate its current power production, the Earth should be even hotter because its surface is much smaller.
