Finny seems almost impervious to fear throughout most of the novel. But, if we look closely at a couple incidents and passages, we can see a couple of his fears surfacing. The main thing that Finny fears is betrayal. He fears what Gene tries to tell him: that the person that he considered his best friend in the world, had purposefully and maliciously crippled him. Finny, an optimist who had given his entire friendship to Gene, won't believe that Gene was capable of such evil. In Finny's world, evil doesn't exist, and to accept Gene's story, is to upset his entire world because it allows for evil and betrayal to exist. When Gene tries to tell him what happened, Finny, in total fear and anger, denies it. He says,
"Of course you didn't do it. You damn fool. Sit down, you damn fool...I'm going to hit you if you don't sit down...I'll kill you if you don't shut up."
Here we see Finny, a normally very serene and go-lucky guy, get really angry, and refuse to believe that his best friend had hurt him. This is Finny's fear manifesting itself. To accept that Gene did it is to accept the fact that evil exists. So, Finny is afraid of evil existing in the world, especially in the form of a friend.
The second thing that Finny fears is incompetence. His entire life, he has had a natural athletic ability, and a natural ability to be liked, accepted, a leader, and the major part of whatever action is going on. When he breaks his leg, he can't play sports anymore, and is rejected from the war. Being rejected from the war was so upsetting to him that he admits,
"Why do you think I kept saying there wasn't any wall all winter? I was going to keep on saying it until two seconds after I got [accepted]."
The war represents the height of the action, and Finny has to be in the center of it. His leg prevents him from being accepted; his leg keeps him from playing the sports that he loves and truly being in his element. These are things that he fears, so he denies them by coming up with an altnernate explanation for the war, and by setting up Gene has his proxy athlete. He fears being incompetent, and a cripple.
Those are just a couple examples of fears that Finny has. I hope they get you started thinking a bit; good luck!