Did the star that the Starkiller Base used ever light up again?

Question

We all know that Starkiller Base used the power of the sun in the system it was currently in, but after it drained that power, did the sun ever come back? Did it re-gain its light?

Answer 1

"Once the star was completely consumed, the weapon was ready to fire": when this happens, the star is dead. From the official Star Wars Inside Intel: Starkiller Base:

Dark energy, abundant in the galaxy, powered Starkiller’s weapon. Harvesting it, however, was not thought possible until the First Order’s success. Using research from the Empire’s Project Celestial Power (the same program that helped create the Death Star), the First Order used the power of a sun to focus and gather dark energy with collector arrays. The First Order had discovered, though, that the most effective way to store the dark energy was by using a planet's natural magnetic field, hence the need to core planets. This, combined with an additional and artificial containment field maintained by the First Order, stored the energy as the weapon charged. Once the star was completely consumed, the weapon was ready to fire.

Then, it would move to another system to reload:

Requiring a sun for each charge of its weapon, the planet-turned-base was capable of moving thanks to thrusters located on the opposite side of the weapon’s massive barrel.

The star death is also explained in the Starkiller Base page of Starwars fandom:

As Starkiller Base was charged through the power of stars, it gradually blocked out sunlight until, running at full capacity, it extinguished it completely, leaving the surface in darkness.

How do stars die?

Stars die because they exhaust their nuclear fuel. The events at the end of a star’s life depend on its mass. Really massive stars use up their hydrogen fuel quickly, but are hot enough to fuse heavier elements such as helium and carbon. Once there is no fuel left, the star collapses and the outer layers explode as a ‘supernova’. What’s left over after a supernova explosion is a ‘neutron star’ – the collapsed core of the star – or, if there’s sufficient mass, a black hole.

What Happens When a Star Dies?

Answer 2

The Starkiller Base is mobile (using hyperspace engines to move its colossal mass) and totally drains its local sun of energy when it's primed. It's described as having 'devoured' it.

The superweapon charges by draining a nearby sun of its power until it disappears. To fire again, the whole base moves to the next star system to devour a new star.

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I think it's fair to say that having all your 'energy' drained will result in the star not being able to light up again, despite that not actually making a lot of sense in our mundane world.