Longer answer: Let's assume you have a "star" (more later) and teleport it into the middle of a Jupiter-like object or other such planet. Jupiter isn't a stars and generally once you get to about thirteen Jupiter masses, fusion begins (for a gas giant) and now it is a star. So, let's go over our stars:
1) Red dwarf: This is the largest "star" object that might fit inside of Jupiter. Some are only 70,000 kilometers in diameter and so could fit inside of Jupiter. Upon teleportation, equilibrium is disturbed and the surface of the red dwarf, pulling upon the gases at hundreds of gs, simply absorbs the gas giant with some alacrity. Likely some stellar brightening, changes in color, for a while.
2) White dwarf: Smaller than a red dwarf, their diameters are only a few thousand kilometers, give or take. You get the same results as a red dwarf but, if the gas giant absorbed has enough matter or the white dwarf is already on the edge, you might get a nova. Not a supernova, that's a different system and a different process. Just a nova.
3) Neutron star: Much smaller than your white dwarf and it will have a larger mass. Again you get a collapse of the gas giant but likely no nova since the gas hits the surface and its electron degeneracy pressure is overcome, blah blah, and everything turns into just another layer of neutrons.
4) Black hole: Not really a star any longer, but some older classifications included it and I will add it for completeness' sake. Much like #3, but instead of neutrons the black hole simply increases in size. You may or may not get an accretion disc, depending on the rotations of your gas giant and your black hole.
So, in all four cases, none of the stars could exist within the gas giant where the gas giant would remain stable.
Longer answer: Let's assume you have a "star" (more later) and teleport it into the middle of a Jupiter-like object or other such planet. Jupiter isn't a stars and generally once you get to about thirteen Jupiter masses, fusion begins (for a gas giant) and now it is a star. So, let's go over our stars:
1) Red dwarf: This is the largest "star" object that might fit inside of Jupiter. Some are only 70,000 kilometers in diameter and so could fit inside of Jupiter. Upon teleportation, equilibrium is disturbed and the surface of the red dwarf, pulling upon the gases at hundreds of gs, simply absorbs the gas giant with some alacrity. Likely some stellar brightening, changes in color, for a while.
2) White dwarf: Smaller than a red dwarf, their diameters are only a few thousand kilometers, give or take. You get the same results as a red dwarf but, if the gas giant absorbed has enough matter or the white dwarf is already on the edge, you might get a nova. Not a supernova, that's a different system and a different process. Just a nova.
3) Neutron star: Much smaller than your white dwarf and it will have a larger mass. Again you get a collapse of the gas giant but likely no nova since the gas hits the surface and its electron degeneracy pressure is overcome, blah blah, and everything turns into just another layer of neutrons.
4) Black hole: Not really a star any longer, but some older classifications included it and I will add it for completeness' sake. Much like #3, but instead of neutrons the black hole simply increases in size. You may or may not get an accretion disc, depending on the rotations of your gas giant and your black hole.
So, in all four cases, none of the stars could exist within the gas giant where the gas giant would remain stable.