Replacing Jupiter with a small star

To add to what Zeiss Ikon said: the barycenter of Jupiter and the Sun together is on average slightly above the surface of the Sun. If all the gas giants lined up, they could cause the barycenter of the solar system to be momentarily further. This means that planets would orbit a point outside the Sun, not the Sun itself. This arrangement has been working well for billions of years.

Jupiter has a mass of 0.0009543 M☉, and it's got more than twice all other giant planets mass combined. The smallest stars have a mass of about 0.08 M☉ (which is still smaller than OGLE-TR-122b, which has 0.1 M☉). That's two orders of magnitude more mass than Jupiter, which makes for that much a stronger pull.

Put that star anywhere within the orbit of Neptune, and the barycenter of the whole system will be WAY outside the Sun. This means that all planets would orbit a point in the empty space between the two stars. If your new star simply pops into existence the orbits would be readjusted in very dangerous ways. In the very least the stars would complete an orbit around the point in very short time (maybe some years, which is almost zero time in geological and astrophysical terms) and in the process they would change distances from each planet enough to destabilize their orbits. Each planet would go into either higher or lower orbits, possibly with most of them colliding against either star and some being thrown out of the system due to gravity assists.

The least massive M class red dwarves (smallest objects correctly called stars) are some 20-30 times Jupiter's mass, even though they're barely larger in diameter. Such a massive object would make any orbits between that secondary star and the Sun unstable, due to the perturbations.

Even a Brown Dwarf (deuterium burner, say) must be at least a dozen or so times Jupiter's mass, so same problem.

If you've read 2010 by Arthur C. Clarke, you may recall that at the point Jupiter was stellified it was by adding a huge amount of synthetic mass in the form of billions (trillions?) of copies of the Monolith. These provided gravity to compress Jupiter to the point of fusion ignition -- but somehow didn't pull everything else nearby off its orbit. This why it's science fiction.