If a star were to suddenly dissapear, would it still have gravity? I am wondering whether or not literally anything can travel faster than the speed of light. For example, if gravity from a star immediately ceases to have an effect if it suddenly and magically vanishes, then this is an example of information traveling faster than the speed of light.
However, if a star's gravity is still present when it disappears, then we have the curious case of things being attracted towards stuff that doesn't exist!
 A: The speed of gravitational waves is finite and very likely the speed of light. If a star (for instance the Sun) suddenly disappeared, we would not immediately feel anything. Assuming the speed of gravitational waves is the speed of light, during more or less eight minutes (in the case of the Sun), we would see the Sun and feel its gravitational field. 
As long as Special Relativity and causality be exact concepts, nothing able to carry information can travel faster than light. 
However, shadows and other things can travel faster than light. You can compute the speed of the spot of a laser that has been aimed at the surface of the Moon and you will realize that can be faster than light, but this spot cannot carry information.
Let me add that thanks to this effect (the finiteness of signal speed propagation), we can study the remote past of the universe (close to its origin) by means of looking at very distant objects. 
A: You can't make the sun disappear. The equations of General Relativity guarantee that the local divergence of the stress tensor is zero. What you can do is convert all the atoms in the sun to gravitons, which rush out at the speed of light. In this case, the gravity is unchanged until the graviton wave hits you.
A: Using the term "gravity" as though it were a material substance that can be given and taken away is conceptually incorrect. All matter has the property of "gravitational attraction" by which it is attracted to all other matter. This property cannot be switched off, so it's not realistic to imagine that it can be. If you're asking whether or not a star's gravitational attraction exists whether or not the star is luminous enough to be visible, the answer is certainly yes. 
