Very interesting question. This is one my students and I discuss every time I teach the novel. There are many different answers to this; just like any work of literature, this novel is subject to many theories. I personally think the two boys are in a codependent relationship, which turns out to be very unhealthy for both of them. I see Gene as very tortured both emotionally and mentally, and Finny as the victim of his madness.
The best textual evidence for my theory of Gene's mental/emotional state is revealed in the first few chapters. It is here that Gene professes his obsession with Finny and his perceived jealousy over Finny's athletic achievements. Read back over these chapters, and note how Gene discusses Finny. First, he praises his roommate, nearly gushing with pride. At one point he says how lucky he is to have Finny as a best friend. However, only paragraphs later, he denounces Finny over some imagined slight. He then revels in paranoia, claiming Finny's been out to get him and destroy his grades the entire time they've known each other.
Many people think that is the reason Gene pushes Finny off the branch: the imagined attack Finny has wrought on Gene's academic legacy. But if you read closely, you'll see that Gene actually realizes Finny was never out to get him before he pushes Finny off the branch. This is perhaps more disturbing, since it means there must have been a completely different motive for Gene's action. What could it possibly be? Only the product of a deeply troubled mind, that's for certain.
After Finny returns to school, the two boys grow even closer. This is where the codependency is obvious. Gene needs to feel like Finny needs him, so he stays as Finny's roommate and seeks to help him any chance he can. Finny needs to live vicariously through Gene; hence the continuous training and exercise. Yet Gene is burdened by his secret, which ultimately destroys not only their friendship, but Finny's life as well.