"Love's Ring Of Fire" (June Carter and Merle Kilgore) was first recorded as a folk song by Anita Carter. It later became a huge pop and country hit, with a new arrangement and some lyric modifications, for Johnny Cash as "Ring Of Fire."
Yes, Elvis's "It's Now Or Never" and Tony Martin's "There's No Tomorrow" are to the tune of the Italian 'O Sole Mio' (which was a Neapolitan street song in the late 19th C.) Not sure if any of those really count as "folk," but in any case it's only the tune they share.
BTW, "The Swallow" (Richard Fariña) is to the tune of the trad. Ladino folk song "Los Bilbilicos," which was also recorded by Theo Bikel.