What are some examples of Finny not hating anyone or anything (other than Gene) in A Separate Peace by John Knowles?
It is true that Phineas (Finny) does not hate people or things in A Separate Peace by John Knowles; you mention Gene as an exception to that, but I'm not so sure he even really hated Gene.
Finny is one of those people who is good at everything, seems to live a charmed life, and whom everyone loves. He is naturally optimistic, in part because he simply refuses to acknowledge any truth which he does not like, and he assumes everyone loves as he does. He believes that
[w]hen you really love something, then it loves you back, in whatever way it has to love.
Finny tells all kinds of stories to get out of trouble, but his stories are relatively harmless and not designed to do anyone any harm. Gene says of him:
Phineas was a poor deceiver, having had no practice.
Again, though, because he does not act deceitfully, he assumes no one else is ever deceitful. When Gene is obviously frustrated and upset about attending the Summer Suicide Suicide Society meetings, Finny does not even recognize it for awhile; when he does, he tells Gene that he does not need to attend the meeting. When Gene is upset because Finny has lured him away from his studies to go with him to the beach, Finny does not even sense it but would certainly have let Gene stay if he had asked. When Gene is secretly seething, Finny does not sense it--and that is not because Gene is a terrific actor. Finny thinks everyone sees the world as he does, assuming the best about people, competing more against oneself than against others, and loving life.
In fact, Gene says Finny would be useless in war because he is so unequipped to deal with anger and hostility. We hear and see Gene's anger and hostility throughout the novel, but Finny displays none of that until his accident--and even then it is minimal until the second incident.
In short, Finny does not have the temperament to hate, even when he has a reason to do so.