Can the Earth leave orbit if the population of humans and other live forms increases?
Life forms are made up from materials already present in Earth. Thus, increasing population would not alter the overall mass of the planet, and can't impact its orbit.
Mass is conserved (up to whatever small contribution nuclear decay has to the overall loss of mass). All biological matter is just created from materials from the environment (we do eat, right?). With exception of an occasional space probe, shooting stars and solar wind effects, the earth can be considered a closed system in terms of matter exchange. Whatever happens on our surface is just a local redistribution of resources.
It's quite easy to compute the energy required to move the earth away from the sun. It's about the same as the kinetic energy of earth around the sun: $2.7\times 10^{33}\,\rm J$. This kind of energy is hard to come by. Planets usually move around and change orbits because other planets have gravitational effect on them. Jupiter was throwing the planets all over the place before the solar system settled in this configuration. But... locally, we can't do anything, because the conservation of momentum holds: INTERNAL forces can't change the mass*velocity product, and mass remains constant, so does the velocity.
As others have said, the mass of the Earth doesn't go up because we are eating food produced on Earth. Suppose we somehow were importing really large amounts of extra-terrestrial food, now what happens?
- We roast. Food contains a fair amount of carbon. Carbon from extra-terrestrial food is just as much a greenhouse problem as burning fossil fuels.
Should we somehow avoid that (say, carbon sequestration) and keep growing:
- We drown. Food consists mostly of carbon, hydrogen and oxygen. The hydrogen and oxygen combine to make water. The sea level rises.
Should we somehow avoid that (say, by importing material to build up the surface with) and keep growing:
- We get crushed. All that mass added to the Earth will raise it's gravity.
Should we somehow avoid that and keep growing:
- We get shaken to death. All that mass being added to the Earth will lower the moon's orbit and thus increase the tides. Eventually this will cause truly catastrophic earthquakes.
Should we somehow avoid that and keep growing:
- We get smashed and buried. As the moon is drawn ever closer it will eventually cross the Roche limit and be destroyed. Our rings would be far more magnificent than Saturn's but there would be an awful lot of mass up there bumping into itself. While much of it just gets ground to dust a decent amount gets deorbited--and even a small portion of the moon is still quite a layer on the surface.
Should we somehow avoid that and keep growing (note that we are now to the point where there will be a slight effect on our orbit--drawing us closer to the sun, not away):
- We go nova. We have a large unburnable core with a bunch of hydrogen (in the form of water) on the surface. That's basically the same thing as a white dwarf collecting hydrogen off it's companion--eventually the hydrogen ignites. (And, yes, our core would be degenerate at this point.)
Should we somehow avoid that and keep growing (and note that we are still orbiting closer, not escaping):
- We turn into a neutron star. There's nothing to hold off the pressure, we cross the limit for electron degeneracy and down we go.
Should we somehow avoid that and keep growing (and note that we are still getting closer, although now if anything the sun is orbiting us):
- We turn into a black hole. Same thing, degeneracy pressure isn't enough and down we go.
Should we somehow avoid that and keep growing (and note that we are still getting closer, although now if anything the sun is orbiting us):
- We fry. Our increasing mass brings the sun down on us just like we brought the moon down several steps ago.
Escaping ceases to mean anything at this point.
I would be very surprised if we could manage to eat enough to even reach the second stage of this list before the aging sun renders the whole matter moot.