I thought there was a FAQ on the topic of "what happens if the sun suddenly disappears" but I can't find it. There is of course the FAQ about the "speed of gravity", which shows that attempting to find the speed of gravity by the direction of gravity is just as wrongheaded as finding the speed of light by the direction of the columb electrostatic force. This approach does not give the correct speed of light - nor does it give the correct speed of gravity. (You may read some popularizations that indicate otherwise. These are mostly the work of one person, Tom Flanderen. These popularizations are wrong).
Details on the "speed of gravity" can be found at
http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html
But back to the original question. There is no solution to Einstein's equations which correspond to the sun suddenly disappearing. The reason why there is no solution can be stated simply, though the deatails are complex: the differential form of the conservation of energy is "built into" Einstein's equations.
Thus you cannot find the solution for what happens when the sun suddenly disappears according to Einstein's equations, because these equations have built into them the assumption that the sun can't suddenly disappear.
Fortunately, there is no reason why it's actually _necessary_ to have the sun suddenly disappear to talk about the idea of the "speed of gravity". One can, in theory, "blow it up", instead of having it disappear. This gives a situation where the equations are soluble.
So the idea is this: one blows up the sun, then, sitting with a stopwatch, one looks for the amount of time it takes for the Earth's orbit to change from it's old predicted value.
There are a few additional wrinkles here. It turns out that in order to measure the speed of gravity, it is necessary to blow up the sun in a manner that is spherically assymetrical. If one blew the sun up symmetrically, the disturbance in gravity would propagate only as fast as the debris traveled - the Earth would not experience any effect in its orbit until the debris actually reached it. (How long this would be would depend on how fast the debris were moving. If they were moving at .1c, it would take about 80 minutes or so for the first of them to reach Earth).
With an assymetrical explosion, one can notice an effect on the orbit much earlier, one that travels at the speed of light. The magnitude of this effect will be small though - it would be determined by the amount of gravity waves emitted by the sun. It would be hard to blow up the sun vigorously enough to generate any significant gravity waves. It would probably be better to make the sun _implode_ rather than to explode it. The fact that the sun is rotating would generate the necessary assymetries, and the gravity wave production would be much higher.
People don't usually actually write out, much less solve, the equations that would be needed to describe these situations, with a few exceptions. The general characteristics of the extremely comlex differential equations that represent GR are already known, and one of the properties that the solutions to these equations have is the property that disturbances in them always propagate at 'c' or less.
Here is the exception - with the development of LIGO, people actually _have_ done the necessary modelling to try and understand what the expected gravitational radiation signals would be from a supernova, or from a binary neutron star inspiral. This is not exactly equivalent to "blowing up" the sun, and is much closer to the case of imploding the sun.
