"What would happen to the solar system if Earth vanished"
Models like these would be amazing to look at, however I know nothing about how hard the implementation would be. Thoughts?
It wouldn't change a lot I think? I think the mass of the sun is so much more heavy than the earth:
PlanetData["Earth", "Mass"]/StarData["Sun", "Mass"]
that the paths of the other planets are hardly affected.
This isn't as solvable of a problem as you probably think it is.
As Sander Huisman pointed out, you don't typically need to include earth in the calculations of the orbits of other planets to get decent predictions well into the future. But if you were to demand to know what the difference is, you'd actually need to calculate a huge, chaotic n-body problem. And if you've ever looked at n-body simulations, you'll know they're not simple by any means.
It'd be instructive to look at just a two planet system and try this. You might find for example that uncertainty in the initial positions and velocities of the planets (due to measurement error) overwhelms the slight difference caused by the deletion of a minor plant in your simulation. I'd claim that it most certainly would have to after a certain point in time.
In short, I would not be confident that what you've requested is possible until an astrophysicist said it was. And even there I'm sure there'd be a ton of caveats about what the simulation "meant".
Not a Wolfram-related suggestion, but if you're interested in such questions you might enjoy Universe Sandbox. I've never played it myself, but from what I hear it's a lot of fun. Not sure about the degree of scientific accuracy though, because as Sean said: These things get complicated quickly, and estimation errors can compound.
as Sean points out this is in general a quite complicated problem. You might want to use special integrations, such as Lie integration, for your simulation. It also depends on how long you need to integrate into the future. Laskar showed using high quality and long term integration that the solar system appears to be unstable (in a certain sense) on long time scales. You might like this preprint; and particular the 2004 paper by Laskar referenced therein.
The way the question is posed I suppose that some understanding of the basic principle is sought. There is this fantastic book, which has very extensive Mathematica code in it. Around page 200 you will find a simplified model of the solar system, with the code to run it. The authors also discuss limitations of that particular model. It is quite straight forward to set the mass of the earth to zero there and see what the model does.
PS: In fact, the entire code is on the website that goes with the book. The code you need is in section 6.2. It creates a diagram for the orbits of all planets (and Pluto). It is relatively easy to remove the earth from the simulation, but you might want to remove it a bit more thoroughly than just setting its mass to zero.