I strongly suspect it is just a mis-spelling of "heart-rending".
In computers, to "render" something is to draw it on a screen or printer, as in "rendering an image". So "heart-rendering" would, I guess, mean to draw a picture of a heart on the screen. :-)
In general, to "render" is to give or present something, like to "render a bill", meaning to give someone your bill for payment. It's barely possible that someone said "heart rendering" meaning "to give someone your heart", i.e. offer love or friendship. But I doubt it, unless they were making a play on words with "heart-rending". As I say, it's probably just a mistake.
The correct modern usage is heart-rending. Rend means to tear. Heart-rending is similar to heart-breaking, an emotional reaction to a very sad event.
You referred to heart-rendering as it appears in an article, December 1861: A heart-rendering scene, written by Dr. Terry L. Jones, a professor of history at the University of Louisiana at Monroe. It is one of a series of articles about the U.S. Civil War. The only use of "heart-rendering" was a direct historical quote from a witness to the 1861 execution (which was then used for the title) in Dr. Jones' article:
"It was heart-rendering," a correspondent wrote, "to see the poor brother's agony."
To render in modern contexts would be:
- In computer graphics, rendering an image
- In cooking, to render fat, the action of rendering fat
EDIT: For example
It was a bad collision, rendering the auto useless.
You could use it. Whether it has ever actually been used is another question entirely.
The meaning of "to render" has been accurately described in the other answers. "To rend" means to rip in two, so "heartrending" is pretty much synonymous with "heartbreaking".
So the quote you gave is probably just a mistake on the correspondent's part (if I may be so bold as to say so), and he meant to say "heart-rending", that is, that it broke his heart to see such agony.
Kenneth Grahame used 'heart-rendering' in his children's classic, The Wind in the Willows (1908). In chapter 2, 'heart-rendering' is used to describe the crash of Toad's caravan in a sudden road accident.
Clearly, 'heart-rendering' was once considered perfectly correct; at the very least, I am willing to believe that Grahame and Charles Scribner II (his publisher) knew their business.
I would like to suggest a slight difference in meaning from the still-common 'heart-rending': while the latter would convey 'heart-breaking', the former might have meant something more sudden and shocking, as with our current 'heart-stopping'. But it would not do to assume that 'heart-rendering' is wrong-headed or incorrect; rather, it appears that that usage has merely become obsolete.
On Language Log, UPenn:
I'm not sure that the people who use heart-rendering are thinking of one of the relevant senses of render, such as "To surrender or relinquish; yield", or "To reduce, convert, or melt down (fat) by heating", as opposed to rend, "To tear or split apart or into pieces violently".
You don't find too many instances of a "mistake" in published literature, see nGram.
About 10,500 occurrences in GoogleBooks, including every possible use of the word combination.
However, I could find no instances on COCA that were not quotes verbatim.
Definitely, some authors use heart-rendering as an eggcorn, some in its possible literal sense (see also LL above, though), some as a mistaken expression in place of heart rending.