Quiescent solar prominences are observed to exist within the solar atmosphere for up to several solar rotations. Their eruption is commonly preceded by a slow increase in height that can last from hours to days. This increase in the prominence height is believed to be due to their host magnetic flux rope transitioning through a series of neighbouring quasi-equilibria before the main loss-of-equilibrium that drives the eruption. Recent work suggests that the removal of prominence mass from a stable, quiescent flux rope is one possible cause for this change in height. However, these conclusions are drawn from observations and are subject to interpretation. Here we present a simple model to quantify the effect of "mass-draining" during the pre-eruptive height-evolution of a solar flux rope. The flux rope is modeled as a line current suspended within a background potential magnetic field. We first show that the inclusion of mass, up to 10 12 kg, can modify the height at which the line current experiences loss-of-equilibrium by up to 14%. Next, we show that the rapid removal of mass prior to the loss-of-equilibrium can allow the height of the flux rope to increase sharply and without upper bound as it approaches its loss-of-equilibrium point. This indicates that the critical height for the loss-of-equilibrium can occur at a range of heights depending explicitly on the amount and evolution of mass within the flux rope. Finally, we demonstrate that for the same amount of drained mass, the effect on the height of the flux rope is up to two order of magnitude larger for quiescent than for active region prominences.