Personally, when considering type match-ups, I like to look at them in terms of either the real world or from a mythological standpoint. It helps to give a basis when considering why something is strong or weak against another type.
That being said, real-world stars are giant spheres of continually combusting gas, while stars take on several different roles in myths and legends. So, I will be considering the Star-type mostly as a real-world star as I make these suggestions, unless otherwise noted.
Normal: I don't believe there needs to be any special interaction with the Normal type.
Fighting: I would either make Star resist Fighting or have a neutral reaction. Even superhuman fighters are often shown to not be able to breath in space. And even if they get to a star, no amount of training would enable them to withstand a star's massive heat output. By the same token, the Sun is a star, and fighters, fictional or otherwise, don't seem affected enough to merit making Star supereffective against Fighting.
Flying: I have less of a scientific basis for this opinion, but I believe Star should be supereffective against Flying and resist it. Mainly, I can't help imagining stars, or any heavenly body really, as "flying" through outer space. Living things of this Earth that cannot fly to outer space. So, to generalize, a star could outfly a bird (Supereffective) and is too far to be effectively attacked by flying (Resistance).
Poison: I would make Star either resist Poison or have a neutral interaction with it. The only thing I can think of for a non-neutral reaction is that, as pollution rips open the hole in the ozone layer, the Sun is able to greatly affect the Earth. Thus, Poison attacking only helps a Star, so Star resists Poison. It's not the strongest support, so I would understand leaving Poison and Star as having no reaction.
Ground: I could understand making Star supereffective against Ground and resisting it. As stars increase in size, they swallow up entire planets (which I associate with Ground), completely vaporizing them. Also, planets don't go crashing into stars because of gravitational forces, which you could (over)simplify into stars repelling (or
resisting) Ground.
Rock: Largely the same argument for Ground, except I wouldn't make Star resist Rock, since meteorites go crashing into other stellar objects all the time. However, if you would identify planets more with the Rock-type than with the Ground-type, I would reverse these two arguments.
Bug: I would make Bug and Star have no special interaction. I will openly admit that this is solely because Bug is already a messed-up type and doesn't need anything else making it worse in my opinion.
Ghost: I would make Ghost and Star have no special interaction. If you associate stars with divine beings, they would be supereffective against mere spirits or ghosts. If you look at it from a "real-world" standpoint, ghosts would just go intangible and be completely unaffected by the star's heat and power. Thus, the two arguments cancel each other out.
Steel: I would make Star supereffective against Steel and resist it. Supereffective for the same reason Fire is supereffective against it. Resist it because the only way humans (or any living thing really) can reach outer space is to build a rocket ship, and even then we have never actually reached a star.
Fire: I would make Star resist Fire. If you generalize a star as a giant ball of fire, a fire attack would just be adding more fire to fire. Similarly, Fire already resists itself, so.
Water: I would make Star super-effective against Water and resist it. I know that seems weird, but I'll explain. Stars are constantly combusting gas, specifically hydrogen and helium. Water is H20, or two hyrdogen atoms and one oxygen. Adding hydrogen to a star would just give it more fuel. Furthermore, in popular culture, whenever the Sun and Earth are depicted as getting close together in terms of an apocalypse, one scene that is often shown is all the oceans evaporating, so the Sun is supereffective against the oceans.
Grass: I have two arguments for Grass.
-- If you associate stars with fire, I would follow the type match-up Grass already has with Fire.
-- If you associate stars with the light they give off, I would make Grass supereffective against Star and make Grass resist Star. This is same reasoning as Grass being supereffective against and resisting Water and Ground. Water and soil are essential for most plants to grow, so using water or ground attacks against a plant only makes the plant stronger. Sunlight would fall into the same category, same reasoning. This is my preferred argument.
Electric: I would make Electric and Star have no special interaction.
Psychic: I would make Psychic and Star have no special interaction.
Ice: Again, I have two arguments for Ice.
-- Associating stars with fire, see the first argument for Grass.
-- Alternatively, I would make Ice resist Star. The empty void of space is extremely cold. Stars seem largely unaffected by anything space can throw at it, so I wouldn't make Ice supereffective against Star. However, even with the massive output of heat from the numerous stars in space, space remains extremely cold. Thus, Ice resists Star. This is my preferred argument.
Dragon: Personally, I would make Dragon supereffective against Star, but I definitely believe Dragon should at least resist it. In several stories, stars come down from the skies and becomes living beings, or vice-versa. However, I cannot recall any stories where a star became a dragon or vice-versa. Thus, whatever creature the star represents, a dragon would trump it. Additional support for Dragon resisting Star is the fact that Dragon already resists Fire.
Dark: I would make Dark and Star have no special interaction. Yes, the light that stars give off is able to travel for millions of miles through the darkness, but there is also so much darkness left unaffected by starlight. Also, viewing Dark as the Evil-type (as it is known in Japan) and viewing stars as celestial beings, the forces of good and evil are constantly battling with neither one having a special advantage over the other. Thus, neutral reaction.
Fairy: My reasoning for Fairy is identical for Dragon.
TL:DR
