pruritus (n.)

1630s, "itching," later, and now exclusively, "having an itching desire for something" (1650s), especially "lascivious, inclined to lewd thoughts," (1746), from Latin prurientem (nominative pruriens), present participle of prurire "to itch; to long for, be wanton," which is perhaps related to pruna "glowing coals" (from PIE root *preus- "to freeze; to burn;" see freeze (v.)).

De Vaan suggests a source in PIE *preus-i-, *prus-no-"(cold and) wet; itching," source also of Welsh rhew, Breton rev, reo "frost," Sanskrit prushnuvanti "to (be)sprinkle, wet," and writes that "The meaning 'to be wet, itch' was metaphorically also applied to high temperatures, hence 'burning' in pruna." But it needn't be metaphorical: Cold damage to skin is called ice burn and the paradoxical sensation of burning when the skin contacts an extreme cold surface has long been noted. Related: Pruriently.