"water from the eye," Old English tear, from earlier teahor, tæhher, from Proto-Germanic *takh-, *tagr- (cf. Old Norse, Old Frisian tar, Old High German zahar, German Zähre, Gothic tagr "tear"), from PIE *dakru-/*draku- (cf. Latin lacrima, Old Latin dacrima, Irish der, Welsh deigr, Greek dakryma). Tear gas first recorded 1917.
"pull apart," Old English teran (class IV strong verb; past tense tær, past participle toren), from Proto-Germanic *teran (cf. Old Saxon terian, Middle Dutch teren "to consume," Old High German zeran "to destroy," German zehren, Gothic ga-tairan "to tear, destroy"), from PIE *der- "tear" (cf. Sanskrit drnati "cleaves, bursts," Greek derein "to flay," Armenian terem "I flay," Old Church Slavonic dera "to burst asunder," Breton darn "piece").
The Old English past tense survived long enough to get into Bible translations as tare before giving place 17c. to tore, which is from the old past participle toren. Sense of "to pull by force" (away from some situation or attachment) is attested from late 13c. To be torn between two things (desires, loyalties, etc.) is from 1871.