25%-50% of all white dwarfs (WDs) host observable and dynamically active remnant planetary systems based on the presence of close-in circumstellar dust and gas and photospheric metal pollution. Currently-accepted theoretical explanations for the origin of this matter include asteroids that survive the star's giant branch evolution at au-scale distances and are subsequently perturbed onto WD-grazing orbits following stellar mass loss. In this work we investigate the tidal disruption of these highly-eccentric (e > 0.98) asteroids as they approach and tidally disrupt around the WD. We analytically compute the disruption timescale and compare the result with fully self-consistent numerical simulations of rubble piles by using the N-body code PKDGRAV. We find that this timescale is highly dependent on the orbit's pericentre and largely independent of its semimajor axis. We establish that spherical asteroids readily break up and form highly eccentric collisionless rings, which do not accrete onto the WD without additional forces such as radiation or sublimation. This finding highlights the critical importance of such forces in the physics of WD planetary systems. ; Comment: Accepted for publication in MNRAS