Why does the father throw out the mother's picture while traveling on the road?
On page 15 of The Road, the father discards all of his former life's identification:
He'd carried his billfold about till it wore a cornershaped hole in his trousers. Then one day he sat by the roadside and took it out and went through the contents. Some money, credit cards. His driver's license. A picture of his wife. He spread everything out on the blacktop. Like gaming cards. He pitched the sweatblackened piece of leather into the woods and sat holding the photograph. Then he laid it down in the road also and then he stood and they went on.
In the post-apocalyptic world, all forms of money, credit, identification, and memories are worthless. Cannibals can't be bribed. Road warriors don't take credit. Even one's spouse is a figment of the past. All that matters is the boy, survival, and carrying the fire.
Also, the mother committed suicide. She gave up on the family. She extinguished her fire. She refused to walk the road. Her photograph is a haunting memory of the father's worst nightmare: that he too will have to use the two bullets in the gun to kill himself and his son. Why carry around that reminder? Her photograph is a constant reminder of self-annihilation. It is best discarded with the other worthless relics of the past.
mstultz72's reply is very good. I would like to build on it by pointing out something in that passage from the novel.
The father is indeed shedding everything from his past that is irrelevant in this new and awful world, but as the quoted passage shows, he doesn't treat all of the items from his past in an identical manner. He spreads out most of the items from his wallet "like gaming cards"; these items are pretty much all the same to him right now. The photograph of his dead wife, however, he holds in his hand. He clearly gives it special attention, sitting down as he holds it, standing up again only after he's placed it on the road along with the other items.
The narator doesn't tell us what's going on in the man's head in this momentary pause in the journey, but we can fill in at least some of the gap here because we see that the image of the man's wife still has meaning for the man. He doesn't part with it as easily as he parts with the rest of the contents of his wallet.