"liquid in a plant," Old English sæp, from Proto-Germanic *sapom (cf. Middle Low German, Middle Dutch, Dutch sap, Old High German saf, German Saft "juice"), from PIE *sapon- (cf. Latin sapere "to taste"), from root *sab- "juice, fluid" (cf. Sanskrit sabar- "sap, milk, nectar").
"simpleton," 1815, probably from earlier sapskull (1735), from sap as a shortened form of sapwood "soft wood between the inner bark and the heartwood," from sap (n.1) + wood, so called because it conducts the sap; cf. sappy.
"weaken or destroy insidiously," 1755, originally "dig a trench toward the enemy's position" (1590s), from Middle French saper, from sappe "spade," from Late Latin sappa "spade" (cf. Italian zappa, Spanish zapa "spade"). The sense of "weaken" probably was influenced by the verb (1725) form of sap (n.1) on the notion of "draining the vital sap from." Related: Sapped; sapping.