Old English borian "to bore through, perforate," from bor "auger," from Proto-Germanic *buron (source also of Old Norse bora, Swedish borra, Old High German boron, Middle Dutch boren, German bohren), from PIE root *bhorh- "hole."
Old English bor "instrument for making holes by boring or turning," from the source of bore (v.1). As "hole made by boring," early 14c. Meaning "cylindrical hole through a tube, gun, etc." is from 1570s; that of "interior diameter of a tube, caliber of a gun" (whether bored or not) is from 1580s. Hence figurative slang full bore (1936) "at maximum speed," from notion of unchoked carburetor on an engine.
1778, "thing which causes ennui or annoyance by dullness;" earlier "state of boredom, fit of listless disgust" (1766); of persons who cause boredom by 1812; usually said to be a figurative extension of bore (v.1) on the notion of "move forward slowly and persistently," as a boring tool does, but OED has doubts and early evidence suggests a French connection.
Le secret d'ennuyer est celui de tout dire (The secret of being a bore is to tell everything) [Voltaire, "Sept Discours en Vers sur l'Homme," 1738]
"be tiresome or dull," 1768, a vogue word c. 1780-81 according to Grose (1785); see bore (n.2). As "cause boredom to," by 1840.