Consider a binary system where gas from one is falling into the other, as is commonly the case when the larger becomes a red giant first.
Use magnetic effects to channel the plasma (if the infalling material is not sufficiently charged, take care of that first) to fall onto one pole, rather than all around the star. That will generate thrust as a jet.
Magnitar — shape the magnetic field to interact with the enclosing galactic field, or with generated magnetic fields. Or, manipulate it to cause a jet to point where you want, and kill the opposite jet.
Any star with a pair of jets, cover one to reflect the thrust. That's essentially the same idea as the shroud posted in another answer, but easier to steer.
Niven & Benford induce a jet in a red dwarf by using a shroud and aim the jet through a hole in the hemisphere.
Generate gravitational waves at different locations and focus them to constructively interfere near the star, making a gravitational gradient that the star falls towards.
Use a "warp drive" to shorten the distance in front and lengthen behind, with the whole solar system in the middle rather than the typical ship. As a plot device, you could have the "payload" required to be a huge mass, so that's the way to travel.
Pseudo-reactionless drive. Convert some stellar material into dark matter, beamed in one direction. It's a jet without the mess. Again, make the density and mass of the stellar core a necessary part of how it works, so it's not just showing off but the way to make it work.
The matrix. Hack the database to change the game board of the simulation we call the universe.
Exotic physics, teleporting momentum. Two stars can essentially repel each other without any standard force between them that anyone would notice.
Given some means to move an exotic object, or that the mechanism would make things inhospitable, you move one star and have it pass near the one you want to move carefully, towing it gravitationally, slingshotting it, perturbing its galactic orbit, or whatnot.
unexplained fictional physics
Any common SF device like inertial dampening fields, shrink rays, tractor beams... can be supersized. E.E. "Doc" Smith had the good guys move planets using the same engine as used for ships: already bought-in to the story's suspension of disbelief, just supersized.