The short answer is NO! In order for a star to ignite its fuel of H to make He , its temperature must reach about 15 million degrees Kelvin, and it is at very high pressure, due to the effects of its huge mass. Now, that mass limit (since mass is what controls the core temperature and pressure due to compression) is about 7% the mass of the Sun. In other words, the smallest star is of mass equal to about 7% of that of the Sun. Jupiter’s mass, BY FAR, the most massive planet is less than 0.1 % of the Sun's mass , nowhere near the limit of 7 per cent!

Then you can see that Jupiter is just too wimpy of a place to ever become a star and ignite the proton -proton chain, and form Helium by nuclear fusion reactions.

Unfortunately Jupiter is too small to ignite. Even the smallest stars that we know about 15 times heavier than Jupiter
massive stars
and these are brown dwarf stars which are different from and dimmer than our own sun.

A huge temperature and pressure is required to start a fusion reaction, and this must be maintained to keep the reaction going. If we could "ignite" a fusion reaction in Jupiter somehow it would just go out because there isn't enough gravity to maintain enough temperature and pressure for fusion.

The planets orbit around the center of mass of the solar system which happens to be the sun because it's so heavy, about 1000 times heavier than the rest of the planets combined. Even if the sun goes out the material is still there and its mass will still be in the center of the solar system so we'll continue to orbit it.