The Earth revolves around the sun because space is bent by the sun, providing the illusion that gravity is causing the Earth and planets to move around the sun whereas they are simply travelling in a straight line within curved space caused by the sun.

If the sun were to instantly disappear, presumably space would return back to its 'non distorted' shape, but the effect would not travel faster than the speed of light. The Earth would, I suspect, continue to travel in its orbit for at least 8.5 minutes, after which it would carry on it a straight-ish line until it fell into more distorted space caused by some other massive body.