What would happen to the solar system if Earth's mass was removed?
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.