Old English tæsan "pluck, pull apart" (fibers of wool, flax, etc.), from West Germanic *taisijanan (cf. Danish tæse, Middle Dutch tesen, Dutch tezen "to draw, pull, scratch," Old High German zeisan "to tease, pick wool").
The original sense is of running thorns through wool or flax to separate, shred, or card the fibers. The figurative sense of "vex, worry, annoy" emerged 1610s. For similar sense development, see heckle. Hairdressing sense is recorded from 1957.