Hearts are certainly important, and were planted throughout the series intentionally:
If you leaf through the series, you'll find either an image of a heart or the word heart in virtually evert issue. Hearts are a major part of what Sandman is about.
Neil Gaiman in The Sandman Companion, part two, chapter 4: "The Doll's House"; emphasis respected.
Does the heart really belong to Koschei? I don't know. Most probably. But is the heart connected to Desire? Almost certainly. As Neil Gaiman acknowledges later in The Sandman Companion, the moral of "The Hunt" is some dreams (and some desires) should stay dreams (see also What did Vassily see when he looked at the sleeping woman? on SFF):
Hy Bender: Any comments about the very end of the story?
Neil Gaiman: Well, the granddaughter thinks the fairy tale's about her and her boyfriend, but of course it's not at all; it's a true story about her grandfather, and about dreams better left unrealized.
Ibid, chapter 8: "Fables and Reflections"; emphasis added.
I would even put forward a theory that this story is similar to "Three Septembers and a January", the story about Emperor Norton, the first and last emperor of the United States. In both stories, Desire tries (in "The Hunt" - subtly, in the other - not so much) to claim the protagonist for its own realm, and on both occasions Dream averts it, by showing them a dream. Norton had a dream of being the emperor; Vassily was granted an opportunity to see the object of his dream, and politely refused it (I wonder if the duke's daughter dreaming of goblets of sour blood had any influence on Vassily).
Although another theory I just developed (after reading this) is that the heart is related to Desire, but through Koschei. It's simple, and confined to this panel:
This is the emerald heart of Koschei the Deathless. He kept his life in this heart, but a woman stole it, and he died.
I.e. here we have the classic story of Desire making someone fall in love and then die because they couldn't be loved back. I think the key difference between Vassily and Koschei in this case is dreams versus desire, respectively. I'm not quite sure where to draw the line between the two in this case, but if we were to accept that the duke's daughter was only a dream for Vassily, then we don't need to involve Desire with another plot to bring a man to his doom.
The choice of words in this panel is what drives my theory:
Notice that is says "dreamed of" instead of "desired". I think the line between dreams and desires in this case is that Vassily didn't truly want the girl to belong to him - he was curious as to whether she truly was as she was depicted in the miniature. Hence he dreamed of her, but without romantic or sexual desire.