Notes in his About section indicate he died without issue, as does The Peerage:
http://www.thepeerage.com/p33544.htm#i335437
Henry Fane was the son of John Fane and Isabel Darell.1 He died circa 1536, without issue.1
Citations
[S37] BP2003 volume 3, page 4134. See link for full details for this source. Hereinafter cited as. [S37]
https://books.google.com/books?id=de09fMymE8AC&pg=PA265&lpg... and several others I've seen state that Henry Vane and Alice Fisher had no issue and I haven't seen a mention of any other marriage for Henry other than to Alice Fisher.
https://www.british-history.ac.uk/survey-kent/vol5/pp177-193
.Henry Fane, the eldest son of John Fane, esq. of Tunbridge, was possessed of it in the reign of king Henry VII. and was sheriff in the 23d year of it. (fn. 4) He died in the 30th year of king Henry VIII. anno 1538, leaving no issue by Alice his wife, sister of John Fisher, gent. of this parish. By his will he gave this manor-place, in which he then dwelt, with all his lands in Hadloe and Capel, to his kinsman Ralph Fane, son of Henry, his father's youngest brother, in tail male, remainder to each of the sons of his youngest brother, John Fane, successively in like tail. (fn. 5)
The last link I posted indicates that Ralph wasn't this Henry's son but the son of his father's brother named Henry:
"By his will he gave this manor-place, in which he then dwelt, with all his lands in Hadloe and Capel, to his kinsman Ralph Fane, son of Henry, his father's youngest brother, in tail male, remainder to each of the sons of his youngest brother, John Fane, successively in like tail. (fn. 5)"