What happened to Handsome Jack?
This wiki entry on Handsome Jack point out that:
He was wearing the mask even before he took control of Hyperion
Which implies to me that whatever the explanation for his face, it would have to do with something about his past that is unrelated to current borderlands events.
Following that line of thinking, the article also offers 3 possible hypotheses:
His daughter Angel did that to him by accident before she could control her powers.
He was branded by a Vault Guardian or Bandit during his search for the vault (before he leaves it to the vault hunters)
His grandmother (who is known to be a Psycho) did that to him when he was younger.
I personally like (3) because of the part in the article and the game when you find out that Jack had his grandmother killed because of some sort of trauma that he underwent at her hands.
The complete article is here.
In addition, I found this video which is an interview by Anthony Burch and Paul Hellquist which says that there is an NPC in Sanctuary who talks about what is under Handsome Jack's mask. Although, they do say that those are just the fan theories that were bandied about on the forums.
Hope this helped a little at least.