Based on what we know from canon, how would Harry breaking an Unbreakable Vow have affected him and the piece of Voldemort's soul inside him? What would have happened?
It was determined in Could the Sorting Hat Have Told Dumbledore That Harry Was a Horcrux that Harry is actually not a proper Horcrux:
JKR: "Here is the thing: for convenience, I had Dumbledore say to Harry, "You were the Horcrux he never meant to make," but I think, by definition, a Horcrux has to be made intentionally. So because Voldemort never went through the grotesque process that I imagine creates a Horcrux with Harry, it was just that he had destabilized his soul so much that it split when he was hit by the backfiring curse. And so this part of it flies off, and attaches to the only living thing in the room. A part of it flees in the very-close-to-death limbo state that Voldemort then goes on and exists in. I suppose it's very close to being a Horcrux, but Harry did not become an evil object. He didn't have curses upon him that the other Horcruxes had. He himself was not contaminated by carrying this bit of parasitic soul."
"I do think that the strict definition of "Horcrux," once I write the [Harry Potter Encyclopedia], will have to be given, and that the definition will be that a receptacle is prepared by Dark Magic to become the receptacle of a fragmented piece of soul, and that that piece of soul was deliberately detached from the master soul to act as a future safeguard, or anchor, to life, and a safeguard against death." J.K. Rowling - Pottercast 12.23.07 - The Leaky Cauldron
So, yes, what would have happened?