11) Ripped apart by tidal forces
You will need: Earthmoving equipment.
Method: When something (like a planet) orbits something else (like the Sun), the closer in it is, the faster it orbits. Mercury, the closest planet to the Sun, moves faster along its path than Earth, which in turn moves faster than Neptune, the furthest planet.
Now, if you move Earth close enough to the Sun, you'll find that it's close enough that the side of the Earth facing the Sun wants to orbit the Sun faster than the side pointing away from it. That causes a strain. Move Earth close enough, within an imaginary boundary called the Roche Limit, and the strain will be great enough to literally tear the planet Earth apart. It'll form one or more rings, much like the rings around Saturn (in fact this may be exactly where Saturn's rings came from). So our method? Move the Earth to within the Sun's Roche limit. Or, better, move it out, to Jupiter.
Moving the Earth out to Jupiter is much the same as moving the Earth in towards the Sun, the most obvious difference being your choice of vectors. However, there is another important consideration, and that is energy. It takes energy to raise or lower an object through a gravity field; it would take energy to propel the Earth into the Sun and it would take energy to propel it into Jupiter. When you do the calculations, Jupiter is actually rather preferable; it takes about 38% less energy.
Alternatively, it may be simpler to move Jupiter to Earth. The theory works like this: build a massive free-standing tower or "candle", with its lower end deep inside Jupiter's depths and its upper end pointing into space. Put machinery inside the tower to pull hydrogen and helium gases in as fuel, through ports in the middle section, and vent these elements out through fusion thrusters at the top and bottom. The tower is called a "candle" because it burns at both ends, see? Now: the flame directed downwards into Jupiter serves to keep the tower afloat (although some secondary thrusters would be needed to also keep it stable and upright). But this lower flame has no direct effect on the Jupiter/candle system as a whole, because all the thrust from the flame is absorbed by Jupiter itself. The two objects are locked together, as if the candle is balanced on a spring or something. The top flame, therefore, can be used to push both the candle and Jupiter along. The top flame pushes the candle which pushes the planet. This is a little unorthodox, and it only works on gas giants, but as means for moving planets it's at least as plausible as the mass-driver and gravity-assist methods described on the earthmoving page.
Earth's final resting place: lumps of heavy elements, torn apart, sinking into the massive cloud layers of Jupiter, never to be seen again.
Feasibility rating: 9/10. As before, impossible at our current technological level, but will be possible one day, and in the meantime, may happen by freak accident if something comes out of nowhere and randomly knocks Earth in precisely the right direction.
Source: Mitchell Porter suggested this method. Daniel T. Staal clued me in on the fusion candle technique, which he got from this Shlock Mercenary comic, which in turn was inspired by the novel "A World Out Of Time" by Larry Niven.