I remember reading some time ago that after the Sun runs out there's only enough energy in a state fit for fusion for like one or two more generations of stars in the universe, and after that the universe will go dark forever.

That may be true, but it's worth noting that the length of a "generation" of stars varies wildly depending on the type of star. Stars like our Sun might only live 10 billion years or so, but red dwarfs live 100 billion years or more so we could presumably live around them long after all the Sun-like stars have burned out.

Can't we dump matter into black holes to draw things out for a while even after that?

From where are you getting the energy to do that?

If you have a bunch of matter (dead stars, planets, interstellar dust) is there some fundamental reason this can't be dumped into accretion disks, etc. in an energy-positive way?

I wonder, does the sun /have/ to run out? Could we find a bunch more hydrogen from outside the solar system and dump it into the sun to recharge it?

Proper management of star lifetime actually requires the opposite approach: subtract mass from the star, so that the star "core" where actual fusion happens can steadily mix with a larger fraction of the remaining mass. The harvested mass can then be stored away as gas-giant planets, and used for miscellaneous purposes.

This is fascinating. Do you have any recommendations for sources so I can learn more about megascale engineering projects to maximize the life of a star?

warning: Isaac Arthur is like crack to sci-fi/intellectuals.

binge at your own risk:

https://www.youtube.com/watch?v=pzuHxL5FD5U

Assuming we possesed that capability it seems like it would be simplet to build a dyson sphere around another star and setup shop on a nearby planet.

Stars fail when the elements in their cores stop releasing net energy when fused. This crossover occurs at iron. Adding more mass will just make it's death more violent.

Also the sun contains 99% of the mass of the solar system. The remaining 1% includes Jupiter. I'm not sure how much effect we can have when only leveraging 1% of 1% of the solar systems mass.

I also wonder how one removes mass from the sun. Its the biggest gravity well in the solar system and our biggest power source is the sun itself.

Star lifting seems plausible.

https://www.youtube.com/watch?v=pzuHxL5FD5U

Transform the mass into energy, and radiate it.

I suspect human will have LONG since destroyed themselves by the time the sun dies, probably before we're spacefaring and out of the solar system.