"injury resulting from violent friction," 1862, from brush (v.2) "move briskly" + burn (n.).
Entries linking to brush-burn
brush v.2
"move briskly" especially past or against something or someone, 1670s, from earlier sense "to hasten, rush" (c. 1400); probably from brush (n.2) on the notion of a horse, etc., passing through dense undergrowth (compare Old French brosser "to dash (through woods or thickets)," and Middle English noun brush "charge, onslaught, encounter," mid-14c.). But brush (n.1) probably has contributed something to it, and OED suggests the English word could be all or partly onomatopoeic. Related: Brushed; brushing.
burn n.
c. 1300, "act or operation of burning," from Old English bryne, from the same source as burn (v.). Until mid-16c. the usual spelling was brenne. Meaning "mark or injury made by burning" is from 1520s. Slow burn is attested by 1938, in reference to U.S. movie actor Edgar Kennedy (1890-1948), who made it his specialty.