It depends on how you remove it. If it was just suddenly not there the short term effects would not be noticed except by us astronomers who wouldn't be able to find it. The long term effects would eventually change the orbits of solar system bodies particularly the asteroids that are closer to it.
However if some alien force applied a "disintergrater ray" to Jupiter we would probably have the effect of a super nova at about less than a light year. We would possibly see a flash an hour or so later and then oblivion. just like the Vorgon fleet did to make way for the space bypass or whatever they called it.
Barry
PS OK I just had to make some sort of a post to make the 300 mark.
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