Dark energy is something predicted by general relativity and iirc it's what is behind the expansion of space (and acceleration of expansion).

Not really. GR can accommodate it just fine, but GR did not predict it. In fact, Einstein, who originally introduced the cosmological constant (which was the term then for what we now call "dark energy") in order to allow a static solution to his equation, thereby missed the chance to predict that the universe was expanding, a decade before Hubble's observations showed that it was. Once it became clear that expanding models matched the data, and did not (then) require a cosmological constant to do so, Einstein dropped it, calling it "the biggest blunder of my life". Only in the 1990s, when evidence for accelerating expansion became available, was the cosmological constant (now called "dark energy") put back into the equations so they would match the data.

> it's what is behind the expansion of space (and acceleration of expansion)

The expansion itself, no. The acceleration, yes.

If you just mean "no dark energy", i.e., the model that most cosmologists used before the 1990s when accelerating expansion was discovered, it wouldn't change the universe's long term fate very much. It would still expand forever and become more and more diluted; the expansion would just slow down forever instead of accelerating forever. There would be some differences as far as our observable universe--we would not eventually become isolated from all other galaxies and unable to see them--but that's about all.