I'd say the human Origin story has the least impact on the game- at least if you mean by "affects the story" to say "adds content." Even if you mean "changes the outcome," this still applies. Highever's fall is a tempest in a teapot. Both dwarf Origins add extra to the experience of returning to Orzammar and the way that section of the story plays out, and the elf ones add extra to the Brecilian Forest or Denerim Alienage "return" experience. The DE even gets a brief surprise visit from his Origin chum- albeit not quite himself. Humans have nowhere to return to, no one to revisit, nowhere that reflects the impact of their Origin experience specially- other than killing Howey, I suppose, adding a few lines before the inevitable deed. Humans do have the unique "be the monarch at the end" option that non-humans don't have, and for many marriage at the end is arguably more impactful then returning to the Origin story location with a unique experience, but I would argue the contrary, particularly if we're talking impact and content from the Origin story itself. If we're talking "makes the ending a lot different" then it's reversed: only the human can make it uniquely different since the impact of every other Origin only really affects the return to the Origin location, not the main story.
I admit I haven't played a human past Lothering yet though, so perhaps I shouldn't assert any of the above...