1540s, "a moral stain;" c. 1600, "a smear on the surface of writing;" perhaps akin to blear. Extended sense of "a confused dimness" is from 1860 [Emerson, in reference to the Orion nebula].
1580s, "blot out by smearing ink over," probably from blur (n.), but the dates are close and either might be the original. From 1610s as "obscure without defacing," also "dim the perception of." From 1856 in intransitive sense "become blurred." Related: Blurred; blurring.