Is gravity's influence instantaneous?

No gravity is not instantaneous. Other wise in the exact case you lay out gravity would have to travel faster then the speed of light, which is impossible.

Believe it or not PBS had a show where the exact same quest was ask use the link below and skip to chapter 3 "A New Picture of gravity" for more information.

Gravity is not instantaneous. Einstein's theory of general relativity says gravity nothing can travel faster than light, and einstein proved that gravity only "acts" at the speed of light. If the sun were pluck out of the universe instantaneous we would continue in orbit for another 8 minutes or so, (the time it would take to see darkness). I highly recomend Brian Green's "The Elegant Universe" book and PBS 3 part series.

Since it is not possible to delete a sun instantaneously, it would be impossible to prove either way, however, gravity is a direct result of mass and if that mass appeared instantly its effect would probably be like a tsunami. The speed at which its effects were felt would be proportional to the amount of mass introduced. It would flow outwardly like a wave and cause a rippling effect for some time until finally settling down to something resembling the equilibrium we see today.

As others have pointed out, gravity travels at the speed of light. In fact, no influence can travel faster than the speed of light. This illustrates the big difference between relativity and Newtonian physics. In Newtonian physics there was instantaneous action at a distance. In relativity the main assumption is that there is a finite speed of causality. This speed happens to be the speed of light.

This is actually quite a fascinating question, and has real importance. Consider: does the earth fall toward where the sun is now, or toward where it was eight minutes ago? Even though gravity waves travel at the speed of light, someone who has crunched the numbers (don't recall who) says that the earth falls toward where the sun is now. Which seems strange indeed.

yes, it would happen immediately. the earth would plunge into cold darkness as it left it's orbit, even though 8 minutes of light would still be radiating from where the sun used to be. depending on the month an day of the year, we could go anywhere. it would be impossible to escape the freezing cold, and life as we know it would just stop. the atmosphere would most likely stay, but it would be impossibly cold and dark. it would be as if life did not exist, oceans freezing over in mere seconds, Niagara falls freezing for the second tme but much quicker and eternally, so if you can make the sun dissapear, please don't :)

I believe that what you are referring to is something known as "gravity waves". Einstein's General Theory of Relativity suggests that changes in a gravational field due to the movement of a body (such as, it is moving away from you) propagate through space at the speed of light (or close to it).

Gravity is absolutely instantaneous - its the attraction of mass - the masses are where they are - they don't just come and go like energy - if a grain of sand moves on a table in front of you its effects are instant - on you me - and even a galaxy 1000 light years away instantly !

the solar radiation made by an explosion in the sun is ejected to the space , and it is not under earth gravity until it is near the planet . So the gravity is instantaneous just with things under his influence

As far as I know, the effects of gravity are limited to c. To reverse your question, if you were to make a new sun appear right next to the old one, it would take eight minutes to feel the pull.