At what energy consumption would we get a 1 degree rise in the Earth's temperature? If energy consumption continues to rise at (say) 4% per year, how long before the heat dissipation seriously impacts climate?
 A: Q=mc(t1-t2)
You need to calculate C, the specific heat capacity of Earth(as a whole).
You need to calculate the specific heat capacity of everything present on, inside earth for that purpose. 
It might be possible after we advance a bit more further:).
A: *

*Global energy consumption is $5\times10^{20}\ J/yr$

*Assume it is all used to power incandescent lightbulbs, so 95% goes to heating the atmosphere

*The mass of the atmosphere is $5\times10^{18}\ kg$

*The heat capacity of air is $1\times10^{3}\frac{J}{kg\cdot °C}$


Assuming all the heat goes to the atmosphere and stays there, using the definition of heat capcity $C$:
$$Q = mC \Delta T$$
$$\Delta T = \frac{Q}{mC}=\frac{\left(95\%\right)\left(5\times10^{20}\ J/yr\right)}{\left(5\times10^{18}\ kg\right)\left(1\times10^{3}\frac{J}{kg\cdot °C}\right)} = 0.095 °C/yr$$
To answer your question about a rise of 4% let's calculate how many years it would take before the atmosphere was heating at 1 °C/yr... oh compound interest formulas, how I've missed you:
$$n=\frac{ln(FV)-ln(PV)}{ln(1+i)}=\frac{ln(1)-ln(0.095)}{ln(1+0.04)}=60$$
In other words, if the rate of energy consumption goes up at 4% per year and nearly all that energy ends up as heat (using lightbulbs as a model), in 60 years the atmosphere would heat up at 1 °C/yr. Please note these calculations are extremely naive and have little bearing on reality in that they ignore 1) factors that actually control the climate, such as gas composition, greenhouse effect, transfer of heat between air-oceans-land, feedback mechanisms, etc. 2) the economics of steady 4% energy consumption increase 3) the fraction of energy consumption that heats the atmosphere. 
A: I suggest to compare human produced heat with the incident heat of the sun which is around 1 kW/m$^2$. The usual comparison is "the sun delivers more heat in an hour than humans use in a day". While such a comparison may not remain accurate forever, a difference in scale of 7000x suggests that even if humans doubled thei energy consumption every 17 years (Thalys roughly 4% per year) it would be w hole before we compete with the sun. This suggests that direct heating of Earth by human activity is insignificant. 
But the extent to which human activity can change (even by a small %) the amount of sunlight captured by the atmosphere is a MUCH bigger deal. If we would cause just 1% more of the sun's power to remain on earth (by slightly modifying the surface properties of Earth or the atmospheric composition) that is a far more powerful multiplier. This is why there is such emphasis on greenhouse gas emissions.
