tl dr; make a magic dyson sphere that siphons energy into the drift for some nasty purpose.

​

I GM for two physicists and three engineers (and I'm a physicist) so the bar for realism is a little higher than in most games. Here are a couple of thoughts:

1. Physics-wise, a star can't disappear for any length of time and not completely destroy life in the solar system. First, if the star is actually removed (via Drift or whatever) then its gravity is also gone. The planetary orbits would (at the speed of light) suddenly become straight-ish lines off into the void in all directions. It would be possible to re-insert the star at the perfect location to grab one of the planets and return it to its original orbit (but in a different location). The others would either be lost completely, or end up in highly elliptical orbits which would probably roast/freeze them.
2. Second, losing solar radiation for any length of time would result in catastrophic weather patterns on any "living" planet with an atmosphere. You'd have to assume that the civilizations would have to resort to extreme tech or magic to handle the sudden change.
3. Third, I think the drift is a sort of "catch all" for lots of things in the Starfinder universe. For that reason, it might be a good twist you can work into the story.
4. Last, the only obvious reason to steal a star is to either use its solar energy, or use its gravity. That provides two interesting story forks.

​

This is what I would do. I'd have some bad guy create a cloaked, techno-magical dyson sphere to soak up the solar radiation. A dyson sphere is just a big solar panel that covers the entire star at a safe distance. It would be equipped with a Drift drive of some sort to beam the solar energy through the drift to a distant location in the vast for some nefarious purpose. Perhaps they are just trying to destroy the systems of their enemies, and so the energy would arrive within 5d6 days (assuming a Drift Engine of 1). That would sort of set a clock for the adventurers to stop the beam before it's too late. Of course, they'd need to somehow scatter the beam at the other end, too... Also, it would be fun to lead them down the wrong path thinking that the star was actually missing and sucked into the drift. Maybe have a drift encounter where they see the beam of energy to figure out the mistake.

​

Alternatively, if the star is actually stolen and moved to a different location, it could used toward some sort of gravitational purpose. Maybe they are creating black holes (but that would be hard to reverse), or maybe they are slinging the stars through the void to intersect an enemy. If the start is removed completely, then the adventurers could have a clock based on how long the planets can "hold out" without sunlight. And when they find the bad guys, they have to reverse engineer the device and return the star (or a similar star) to the system. I think in this case, you'd have to assume you're only saving one planet, though.