Ok
well lets completely ignore the logistics of making mass dissapear for a moment and do a quick consideration of the centre of mass of the planet/planet or planet/sun system.
COM = (M1*r1+M2*r2)/(M1+M2)
Lets do this for the Jupiter/Sun system as it will probably have the largest effect (upon removing).
COM = (1Ms*0+0.000955Ms*(7.78*10^11))/1.000955Ms
Here I have put the mass of jupiter in terms of the sun and placed the sun at the origin. This gives the jupiter sun system a centre of mass of ~740 thousand km from the centre of the sun. This may sound like a lot, but the sun has a radius of 1.4 million km, so the centre of mass is well inside the sun.
I would suggest then that removing jupiter would not have much effect of the sun - but it would certainly have a large effect on the asteroid belt, probably sending a lot of rocks into eccentric orbits..