If the wormhole travels all the way to the core, then it will relocate about half of the star to the other end. If the wormhole can be closed, then it would act as a giant pizza cutter for stars. Potentially very useful to defuse supernovas. Or to utterly annihilate your enemy's stronghold...

What would happen is more complicated to answer if the wormhole stays open. The star would effectively have double the surface area, but apart from that, the consequences for the star's future development elude me right now.