means A drop of clear salty liquid secreted by the eye, shed in crying or to keep the eye moist — or, said another way, to rip something apart by force.

from The two meanings are two different words wearing the same spelling. The eye-drop "tear" comes from Old English 'tēar', related to Old Norse 'tār' and a wide Indo-European family that also gave Latin 'lacrima' and Greek 'dákry'. The rip-it-apart "tear" comes from Old English 'teran', a cousin of German 'zerren' (to tug) and rooted in a sense of pulling and pluck­ing apart. English just happens to spell both the same, so the eye and the gesture of ripping share a page by accident — though there's a quiet poetry in it.