I would say that this quote, which means that a child should not be punished for his or her father's sins, does also apply to Cory, but in a rather different way. Throughout the play we see Cory dominated by the overbearing presence and ideas of his father, and he does suffer under his parent's shadow. At the end of the play, after Troy's death, he remembers his father as a constant grim, hovering presence as he was growing up, a kind of spectre almost, that haunted him continually and made him afraid:
The whole time I was growing up ... living in his house ... Papa was like a shadow that followed you everywhere. It weighed on you and sunk into your flesh. It would wrap around you and lay there until you couldn't tell which one was you anymore.
We see from the close of this quote that Cory felt so cowed and dominated by his father that he was left unsure of his very identity, his own individuality. We can conclude therefore that Tory inflicted a certain amount of psychological damage on his son, most notably in denying him a football career simply because he didn’t believe it was the right path for him.
However, it should be noted that, although Cory is in this sense punished for Tory’s faults, Troy’s domination of his son is not really presented as a major sin in this play. Rather, Troy’s interference in his son’s life is seen to result from his over-protective nature. He hurts Cory, but he does not mean to. He just does not want Cory to have the kind of hard life that he himself had when growing up. He is trying to shield Cory from the harshness of the world but the trouble is he goes about it in entirely the wrong way, trying too hard to impose his own ideas and not stopping to consider what his son might want. This inevitably alienates Cory from him.
Cory, then, feels stifled by his father; and although Troy is actually well-intentioned towards his son, his domineering behaviour renders this a rather unfair and unequal relationship. However, Cory is also seen to be capable of breaking away from his father’s influence when he leaves home and finds his own way in the world. It does not seem likely that Cory will suffer any more from the failings of his father. In any case, after Troy's death, although at first he declares he won’t go to his father’s funeral, in the end he does so. This suggests that he is able to come to terms with his past, after all, and that it will no longer haunt him.