What would happen to the solar system if Earth's mass was removed? This question is inspired by the "Stargate SG-1" episode "The Road Not Taken". In that story a device is used to shift the entire Earth to another dimension to defend against an attack.

I'm not asking about the science of whether that is even theoretically possible but rather the effect on the rest of the objects in our solar system if it did happen. Since this was apparently a short shift (likely days at the most) I would imagine there might be little overall effect to most things but it occurred to me the Moon might immediately start drastically altering its position relative to the now massless Earth (there was no indication that the effect extended to the Moon but given that ships very close to Earth were unaffected it would seen that the Moon was also unaffected) so that when the Earth returns to "normal" it may no longer be in Earth's orbit or worse, Earth may rematerialize partially within or completely around it.
There would be three effects to consider, one very short-term, one medium term and one long-term.

The medium-term effect is that the Moon has an orbital velocity around the Earth of about 1 km/s.  If the Earth suddenly disappeared, the Moon would just keep on going in a straight line at 1 km/s.  This would have the effect of putting it into a Solar orbit quite different from the Earth's. It would have a different inclination to the ecliptic (the Moon's orbit is inclined to the ecliptic by 5 degrees), different period and different eccentricity. (How different depends on exactly where in the Moon's monthly orbit the Earth disappeared.) 

The effect wouldn't be enough to cause the orbit to intersect that of Venus or Mars right away, but perturbations on the new Planet Luna would potentially be very different from those on Earth and it might evolve fairly quickly (centuries) to a Mars-crossing or Venus-crossing orbit with an eventual collision.

It is highly unlikely that it would be very near the spot where Earth reappears.

The long-term effect is smaller: The Earth is part of the generally stable Solar System and contributes small perturbations to all the planets. It is possible (but I suspect unlikely) that the disappearance of the Earth and the subsequent lack of the Earth's perturbations would result in significant changes to the Solar System over many centuries to tens of thousands of years.

The very short-term effect is that the disappearance of the Earth's mass would result in a speed-of-light propagation of an abrupt change in the curvature of spacetime.  I.e., gravitational waves.  I don't know enough to calculate if these waves would have any noticeable effect within the Solar System. My inexpert guess is that it's plausible that they would.