I'm not sure what you mean by "linear expansion". If you mean that the rate of change of the scale factor with respect to time never changes at all, that is not possible except in the edge case of a universe containing zero matter or energy and zero cosmological constant. (This edge case is often called the Milne universe.) Which is obviously not the universe we live in.

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.