All we know for certain is the the phrase was used in Vanity Fair (NY) in 1927. Singers of smoldering music were called "torch singers" during in 1920s.
We probably won't know the answer to this until the famous New York humor magazines Life, Puck, and Judge are digitized. That probably won't be soon, so don't hold your breath (or a torch).
(Oxford English Dictionary)
fig. or allusively. Something figured as a source of illumination, enlightenment, or guidance, or of heat or 'conflagration'. Also in phrs.: to hand (pass, etc.) on the torch (and varr.), to pass on a tradition, etc., esp. one of enlightenment (after L. lampada tradere, Gr. , a metaphor from the ancient Greek torch-race; cf. LAMP n.1 1c and see sense 3 below); to carry (etc.) a torch for (someone), to feel (esp. unrequited) love for, to feel lingering affection for.
1621 BURTON Anat. Mel. III. ii. VI. i. (1651) 545, I light my Candle from their Torches. 1664 JASZ-BERENYI (title) A new Torch to the Latine Tongue. 1775 SHERIDAN Rivals Epil., The torch of love. 1878 R. B. SMITH Carthage 19 The torch of Greek learning and civilisation was to be extinguished. 1887 Q. Rev. Oct. 276 Her [sc. Italy's] work has been done among the nations, and in their turn France, England and Germany hand on the torch. 1912 E. GOSSE Portr. & Sk. p. viii, They were all..engaged in keeping bright, and in handing on unquenched, the torch of literary tradition. 1927 Vanity Fair (N.Y.) Nov. 132/3 When a fellow 'carries the torch' it doesn't imply that he is 'lit up' or drunk, but girl-less. His steady has quit him for another or he is lonesome for her. 1932 L. GOLDING Magnolia Street I. xi. 189 He had sometimes hoped that in Max a son was born to him who would take the torch from his dying hand and jump on to the platform he had vacated. 1953 L. Z. HOBSON Celebrity vi. 78 Jim's still carrying a torch for Roosevelt. 1959 Manch. Guardian 16 June 5/2 She was carrying a torch for someone. 1969 J. GROSS Rise & Fall Man of Lett. iv. 104 Dante was the poet of Catholicism, who handed over the torch to Shakespeare, the poet of Feudalism, who passed it on to Milton, the poet of Protestantism. 1977 H. FAST Immigrants v. 305 Maybe you got a torch for her, maybe not. But we both got her interest at heart.
1 March 1935, Washington Post, "Broadway" by Ed Sullivan, pg. 12:
Nick Long, jr., is carrying a terrific torch for one of the chorines in "Say When."
6 October 1936, Ironwood (MI) Daily Globe, "In New York" by George Ross, pg. 4 ,col. 7:
"Carrying the torch" is now - via Hollywood and the radio - familiar to everyone but it's one of Broadway's oldest slang terms.