Yannis essentially answers the question here, but I'll add a little detail, just for fun, using poetry (since the deepest insights come from great poets) and linguistics.
T.S. Eliot writes about this very specifically:
Who then devised the torment? Love.
Love is the unfamiliar Name
Behind the hands that wove
The intolerable shirt of flame
Which human power cannot remove.
We only live, only suspire
Consumed by either fire or fire.
Little Gidding
The previous stanza elucidates further:
The dove descending breaks the air
With flame of incandescent terror
Of which the tongues declare
The one dischage from sin and error.
The only hope, or else despair
Lies in the choice of pyre of pyre-
To be redeemed from fire by fire.
Little Gidding
The Ancient Greek word for fire is "pur" (πῦρ) from which we derive pyre and purify.
In the Four Quartets and other works, Eliot connects these ideas from the Upanishads, through Greek Mythology and into Christian lore. Fire, is, of course, very important in Hindu mythology, and the Christian conception of love as agape grows out of ancient Greek explorations of the deeper meaning of love beyond eros. Eliot purposely conflates eros and agape in this poem.
It is the romantic jealousy of Nessus that "weaves" the poison shirt, Deianira's love for her husband that brings it to him, but in the Christian conception, it is love that redeems us from the inextricable suffering of existence.
It's not clear that Heracles can die--part of his myth is his defeat of death (Alcestis), and his travel to and return from the underworld. Jumping off a cliff or opening his veins or falling on his sword may not have worked.
But by burning himself alive, he experiences suffering so excruciating that Zeus takes pity of him and releases him from the mortal coil.
Pity was a very important concept to the Greeks. It humanizes, as in the death of Hector, breaker of horses. It was deeply religious, in that catharsis was the intent of Greek drama.
The purification and burning away of the mortal flesh can be said to be the foremost meaning, but pity is the key.
And all shall be well and
All manner of thing shall be well
When the tongues of flame are in-folded
Into the crowned knot of fire
And the fire and the rose are one.
Little Gidding