How do Phineas and Gene grow throughout the novel?
Gene and Finny are roommates in A Separate Peace, and your question asks how they grow. Growth implies moving forward, and I'm not so sure either of them makes forward progress in this novel; however, they do each undergo a dramatic change. Phineas, once an eternal optimist and idealist who saw the war (and, frankly, everything else) as some kind of a cosmic game, is transformed into a bitter realist. He still claims the war is a giant fabrication created by old men, but he doesn't really believe it anymore. The change, of course, is precipitated by the reality of Gene's disloyalty and (though he doesn't want to admit it) momentary hatred.
Gene is more of a realist, committed to achieving and producing, though he loves neither. He is a friend to Finny, though Finny is a better friend than Gene. Gene gets frustrated at Finny for all the distractions and the dares (jumping from the tree) and the devilish antics which keep Gene from moving forward on his plan of achievement. Somehow Gene determines that Finny is deliberately trying to sabotage him for his own gain; it's not true, but in a moment of anger he jounced the limb which ultimately leads to Finny's death. Nearly the moment that happens, Gene realizes his mistake, but it's too late. His outward change, then, is from friend to enemy and back to friend, though damaged and broken. Internally, Gene goes from believer to cynic, back to believer then back to cynic. There are lots of ways to phrase that journey, but that's essentially it. Gene is a more complicated character, of course, because we meet him many years after the event, as well. As an adult, he can be read as having changed and grown from his experiences at Devon; or he can be seen as having undergone no change at all, still cynical and still rather angry.
If loss of innocence is growth, these two characters grow---a lot.
I will assume that you are asking individually as characters, not corporately as a relationship.
Finny: In the beginning, Finny's competitive sporting spirit makes him fun-loving but also determined to be better in one way than everyone else. After the accident, he obviously can't express this portion of himself in any way, even though he tries through Gene. This makes him have to consider what else he has to offer. We see this as when he writes letters across the world trying to secretly participate in any army that will take him. His growth is demonstrated in the jobs he was willing to work: desk jobs. This is not the Finny we met in the beginning.
Gene: Gene is a great character to watch because we feel his pain as he narrates. Who hasn't betrayed a friend and then felt bad about it. Gene's greatest redeeming moments are often unknown to the other characters. For example, after Finny's second accident, Gene rushed to the infirmary to be with him but he gets refused. Accepting Finny's feelings is another example of demonstrating growth. Teens especially try to make everything right, even when it's not.