Micropipet aspiration of phase-separated lipid bilayer vesicles can elucidate physicochemical aspects of membrane fluid phase coexistence. Recently, we investigated the composition dependence of line tension at the boundary between liquid-ordered and liquid-disordered phases of giant unilamellar vesicles obtained from ternary lipid mixtures using this approach. Here we examine mechanical equilibria and stability of dumbbell-shaped vesicles deformed by line tension. We present a relationship between the pipet aspiration pressure and the aspiration length in vesicles with two coexisting phases. Using a strikingly simple mechanical model for the free energy of the vesicle, we predict a relation that is in almost quantitative agreement with experiment. The model considers the vesicle free energy to be proportional to line tension and assumes that the vesicle volume, domain area fraction, and total area are conserved during aspiration. We also examine a mechanical instability encountered when releasing a vesicle from the pipet. We find that this releasing instability is observed within the framework of our model that predicts a change of the compressibility of a pipet-aspirated membrane cylinder from positive (i.e., stable) to negative (unstable) values, at the experimental instability. The model furthermore includes an aspiration instability that has also previously been experimentally described. Our method of studying micropipet-induced shape transitions in giant vesicles with fluid domains could be useful for investigating vesicle shape transitions modulated by bending stiffness and line tension.