Earth radiates heat away at a constant rate only because Earth's temperature is approximately constant. Actual black-body radiation is governed by the Stefan-Boltzmann law [1], and is proportional to temperature^4. A small increase in temperature significantly increases the amount of energy radiated, which then serves to moderate the temperature.
In practice, the total energy consumption of the Earth is bounded by the amount of solar energy that falls on the earth, which is about 100,000 times greater than current total world energy consumption. For what civilizations might look like that exceed that, check out the Kardashev scale [2], which classifies civilizations by the size of the astronomical object whose energy they harness. Type 1 civilizations harness a planet, Type 2 harness a star, Type 3 harness a galaxy. We're a lowly Type 0 civilization, which means that we have not yet attained the technology needed to make full use out of all of our planet's resources.
[1] https://en.wikipedia.org/wiki/Black_body#Radiative_cooling
[2] https://en.wikipedia.org/wiki/Kardashev_scale