What is the origin of the idiom/phrase "Heroes never die"?

The earliest-published Google Books match that I could find for any of the various versions of the expression is from the epilogue ("by Walsh Porter, Esq.") to William Dimond, Hero of the North: An Historical Play (1803):

But we have Heros, it must be confess'd,

From South as well as North, from East and West—

Long, long, I trust, to grasp th' immortal prize,

For he, who lives a Hero, never dies!

An early repudiation of the same idea appears in a couplet translated by the Rev. E. Stokes at an unknown date and published in Henry Wellesley, Anthologia Polyglotta: A Selection of Versions in Various Languages, Chiefly from the Greek Anthology (1849):

False is the tale ; a Hero never dies.

Or Alexander lives, or Phœbus lies.

The Greek original of this epigram is attributed to Parmenion, whom Wikipedia identifies as the (third-century BC) architect employed by Alexander the Great to build Alexandria. And since he views the "tale" that a hero never dies as false, that means the tale was abroad before the rejoinder.

The Anthologia Polyglotta includes a Latin version of Parmenion's epigram, too; Wellesley credits that translation to Hugo Grotius, who died in 1645.

In any case, by conservative estimate, the sentiment that heroes never die and the counterclaim that this sentiment is false have been in the world for at least 2300 years—and available for translation into English for as long as there have been English-speaking people capable of finding and reading the Greek Anthology.