Earth orbits the sun with a velocity of $29.78 km/s$. In order to fall into the sun that velocity would have to be reduced to approximately zero. The reason this velocity would have to be reduced to zero is an energy balance. Orbits are constantly balancing kinetic energy with gravitational potential energy, if we slow our velocity, we'll fall slightyl toward the sun picking up a bit of speed, that would let us orbit at our new distance from the sun. the most efficient orbital transfers apply two trusts one at the beginning to start the transfer to the new orbit, and one at the end to stabilize it. We wouldn't need the one at the end, but in order to get an orbital transfer that intersects with our sun our velocity would have to drop to near zero.  So that's the $\Delta V$ for our [rocket equation][1]. If we used our best rockets we can get an exhaust velocity of about $4.4 km/s$. We would have to use 99.9% of earths mass as reaction fuel and the remaining 0.1% of earth would fall into the sun. If we were really dedicated to the cause we could try to use our most advanced ion thrusters with an [*exhaust* velocity][2] of up to around $210 km/s$. This would bring down our fuel mass to a more reasonable 14% of earth. However, the time-span required to consume 14% of earth one proton at a time would be enormous. But I suppose that's good, because the energy required to do so would take more than all of the *current* chemical energy on the planet, so we'd probably have to use solar power and just wait for more energy to arrive from the sun.

So in theory, even if we were *trying* to send earth into the sun, it would take a herculean effort and we wouldn't even make a dent by the time anyone reading this dies.


  [1]: http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation
  [2]: http://en.wikipedia.org/wiki/Specific_impulse#Examples