Sadly, a very small one. In the post-apocalyptic world in which the father and the son are trying to survive, compassion doesn't have much room, and could actually be considered a survival threat. If one shows too much compassion, then your own chances of survival are depleted, whether through resources, or through trusting people that might turn out to be dangerous. We see compassion mainly in the small boy--any time they run across a surivor, he has a powerful and compelling urge to help them. While this is an admirable human trait, that would not doubt fall under the symbolic "fire" that the father wants his son to carry at all times, it is an unrealistic, dangerous, and often fruitless urge.
One of the first survivors they see is the man who had been struck by lightning; no help they could have given would have done any good. The boy shows compassion anyway, and wants to help. Later, they encounter a very old man, and do try to help him a bit, but don't have enough resources, and the old man is wary and suspicious of questioning and friendship. Prompted by the boy's compassion, they do what they can, and then leave.
The other instance of compassion that is seen, other than in the boy (and in the father's attempts to teach the boy as best he can), is in the man who finds the boy in the end. We are left assuming the man takes the boy into his own family; this is a huge act of compassion that is highly risky for this man, but he does it anyway. Other than that, this story is sadly devoid of much compassion--a lack of it was what probably helped to destroy the world, and not much of it was left afterwards. I hope that those thoughts helped; good luck!