From the Online Etymological Dictionary:
The oft-heard statement that male homosexuals were called faggots in reference to their being burned at the stake is an etymological urban legend. Burning was sometimes a punishment meted out to homosexuals in Christian Europe (on the suggestion of the Biblical fate of Sodom and Gomorrah), but in England, where parliament had made homosexuality a capital offense in 1533, hanging was the method prescribed. Any use of faggot in connection with public executions had long become an English historical obscurity by the time the word began to be used for "male homosexual" in 20th century American slang, whereas the contemptuous slang word for "woman" (and the other possible sources or influences listed here) was in active use. It was used in this sense in early 20c. by D.H. Lawrence and James Joyce, among others.
I think the use of the word is related but only indirectly to the burning of heretics. In the 14th century the word was used to refer to a bundle of sticks used as firewood. By the 16th century the word was used in reference first to wood used in the burning of heretics. In the 17th century reformed heretics wore the emblem of a faggot on their sleeves to symbolize their recantation, and eventually the word came to be used in reference to the heretics themselves. An 1868 use of the word illusrates this: “Wolsey caused them to carry a faggot to the fire. Henry placed them in the midst of actual faggots.” In the 19th century the word was used in England as a term of contempt for a woman, and in the early 20th century it began to be used in the US as a term of contempt for homosexuals.
My understanding is that the connection between burning heretics and homosexual reference is separated by several generations of development. I think the most reasonable interpretation is that folks in the US borrowed a deragatory British term for a woman and used it to refer to homosexuals. The immediate connection is femininity, not heresy or fire. It is the same sense that the term sissy derives from sister.