"Now, if you want to get BILLIONS of people into space (and the infrastructure that they need to survive), and you want to do it fast enough to beat population growth on Earth, then you've got to burn chemical energy tens or hundreds of times faster than the whole world is doing at the moment."

No, this wrong- in a number of different ways.
The volume of space launches each year is an important factor in determining the means in which we get to space.
So if there are thousands or million of launches per year, other technology which is other than chemical rockets will be used.
But even assuming it's done the same way with chemical rocket, and assuming 200 million people per year- so in a decade that is 2 billion less and within 4 or 5 decades hardly anyone would left on Earth.
And assuming a generous amount of 2000 gallons of kerosene per passenger, times 200 million gives 400 billion gallon of kerosene
The US uses 146 billion gallons of gasoline per year.
http://auto.howstuffworks.com/question417.htm

If of the 200 million people leaving earth each year 150 million are from the US in the first year [and that's all that leaves the US] there should be about 73 billion gallons less used each year, so in 10 years it "pays" in fuel use for about 2 years of people leaving. And less than 3 decades just counting the US, it would "pay" the total fuel cost of billions leaving Earth.
Now, if the only fuel used is Kerosene you might get some shortages of Kerosene. You could use Hydrogen. And you could get energy from space, that is then used to make the Hydrogen from water. And when hydrogen is then burned it makes water. So it's possible that no energy is used from earth sources, and there is no pollution or production of CO2.
But in any case it's not burning "chemical energy tens or hundreds of times faster than the whole world is doing at the moment". It might equal to amount of all chemical energy [coal, natural gas, gasoline, ect] used by the US in about a year per each year of launching 100-200 million per year.