What would happen to the universe if dark energy started "disappearing"?

If we take the present-day universe, adopt the so-called concordance cosmology, or Lambda-CDM model, accept GR and the Robertson-Walker metric, but then allow the dark energy content to be suddenly zero, then the universe will no longer be flat. Instead it will have an open geometry and the energy density within it would be dominated by dark matter. However, this energy density would have insufficient gravitational influence to halt the current expansion. As the density universe get bigger, the density falls and the influence of gravity becomes even weaker.

$Omega_M simeq 0.32$, where $Omega_M$ is defined as the ratio of the matter density to that required in order to make the universe flat. Thus, although the expansion would decelerate, matter (including dark matter) is short by a factor of three of being able to halt the expansion at any point in the future. Only if $Omega_M geq 1$ does the expansion come to a halt. These solutions were explored by Friedmann in 1922. The plot below (taken from this wikipedia page) shows solutions with $Omega_M = 0.3$, which can be contrasted with the concordance model of $Omega_M simeq 0.3$, $Omega_{Lambda} simeq 0.7$, which is also shown as an accelerating expansion.

To give any other answer would require you to be a bit more definite on what you mean by "not simply disappearing".