Mountains are distributed across the surface of Io, the fiercely tidally heated moon of Jupiter. The large crustal thicknesses implied by their great heights can be reconciled with Io's high heat flow, if most of the heat escapes directly via volcanic eruptions (the heat-pipe model), but the origin of the mountains has remained obscure. Recent images show that many of Io's mountains are tilted blocks undergoing tectonic collapse, and we propose here that the volcanic heat-pipe (and continuous terrain burial) model naturally leads to such unstable topography. That is, burial (1) generates horizontal tensile stresses as the volcanic crustal stack is loaded, (2) creates large horizontal compressive confining stresses as Io's crust subsides (moves to a smaller effective radius), and importantly, (3) allows for potentially large horizontal compressive thermal stresses as the base of the crust reheats owing to fluctuations in the efficiency of the volcanic heat piping. Faulting associated with these stresses may raise mountain scarps directly or in concert with thermal uplift due to the crustal reheating; continued crustal heating and melting then lead to mountain collapse (all over <1 m.y. to a few million years). Our model predicts that regions of active mountain formation and volcanic activity on Io should be anticorrelated, which is observed. Moreover, substantial tidal heating and disruption of planetary crust are seen elsewhere in the Jupiter system, in the chaos terrains of Europa. There may be stronger commonalities between the two inner jovian moons (and early Earth) than previously realized.