Small, rocky planets have been found orbiting in extreme proximity to their host stars, sometimes down to only ~2 stellar radii. It is generally agreed that these ultra-short-period planets (USPs) cannot have formed in their present-day orbits and must have migrated from larger separations, likely due to tidal dissipation. The primary source of this dissipation has remained uncertain. Here we show that planetary obliquity tides naturally produce USPs through runaway migration of the innermost planet of some close-in multi-planet systems. Specifically, the end state of tidal dissipation (known as a “Cassini state”) is a planetary obliquity (axial tilt) that is in general non-zero, sometimes significantly so. This is unlike the eccentricity, which tidally equilibrates at e=0. When the forced obliquity is non-zero, sustained tidal dissipation is inevitable. If a planet’s initial semi-major axis is small enough, the associated tidal migration leads to continued obliquity excitation and runaway orbital decay, which is stalled when the forced obliquity reaches very high values (~85 deg) and becomes unstable. We outline the conditions in which the innermost member of a prototypical Kepler multi-planet system becomes a USP. The mechanism requires little fine-tuning and can simultaneously account for empirical trends with stellar mass and period ratio.