Earlier quoted context omitted.
That "trick" would indeed work and is a well known orbital manoeuvre: http://en.wikipedia.org/wiki/Bi-elliptic_transfer (without the final circularising burn). This manoeuvre would be preferable to the article's alternative (a Hohmann transfer reducing periapsis to the sun's radius without the final circularising burn) in all cases where the initial semi-major axis (relative to the sun) is roughly greater than 11.94…
Sorry, your link doesn't mean what you think it means. What you're missing is that both waste and the earth are already in orbit around the sun whereas the maneuver your link describes involves switching between orbits around different smaller bodies. Edit: Which is to say as the other have mentioned, this doesn't send the waste into the sun (as the parent post implied) but merely on some orbit similar to the earth a…
A bi-elliptic transfer is used for transferring between two orbits around the same body. In some situations, this is more efficient than a straightforward Hohmann transfer, at the cost of taking much, much longer.
For a Hohmann transfer, you reduce the ship's orbital velocity relative to the sun, reducing the orbit's perihelion to somewhere inside the sun. That requires cancelling out almost all of the Earth's orbital velocity.
For a bi-elliptic transfer, you first increase the ship's orbital velocity relative to the sun, increasing the orbit's aphelion to somewhere in the outer solar system. That's your first Hohmann transfer, going apparently the wrong direction, but requires less than 40% as much energy as dropping it into the sun did. Then, at aphelion, when the ship's orbital velocity is lowest, you reduce the ship's orbital velocity to drop it's perihelion inside the sun. From that far out, and at such an eccentric orbit, that requires only a fraction of the energy it did from Earth's orbit.
The entire manuever, potentially, could be far more efficient than a direct transfer. It just takes far longer, and if the ship fails to make the second maneuver, it'll be left in an orbit that crosses Earth's orbit.