This is a difficult question, in that Finny is the truly innocent character of the novel, while Gene harbors paranoia & jealousy which lead to Finny's fall and, ultimately, death. However, Gene does carry some innocence of adolescence with him early on.
One such example is his attitude toward war, and his status in society as a 16-year-old male in the summer of 1942.
"I think we reminded them of what peace was like, we boys of sixteen. We registered with no draft board, we had taken no physical examinations. We were carefree and wild, and I suppose we could be thought of as a sign of the life the war was being fought to preserve. Anyway, they were more indulgent toward us than at any other time."
Gene has the advantage of being a year younger than those who are preparing for war, & this allows him a carelessness & selfishness which not many people could possess. He is free to worry about his trigonometry test on the night he & Finny sleep on the beach, & he is free to jump out the tree as a game, rather than preparation for fleeing a sinking ship. They are able to complain about lack of maids to make their beds, rather than lack of nurses to tend their wounds on the battlefield. He can pursue the goal of valedictorian, instead of choosing which branch of the military to join.
Gene, in many places, also represents an innocence in friendship. Despite his animosity toward Finny, and his paranoia in thinking Finny is out to sabotage his grades, Gene often delights in realizing the depth of their friendship. He is excited to know that the most popular kid in school is his roommate and has chosen him to join in all his ridiculous antics. These moments highlight how dramatic the shift in his thinking is, when it comes later.
Perhaps the descriptive adjective applied to Gene in the early part of the novel can be ingenuous rather than innocent, an adjective whose connotation differs in that it suggests a lack of guile rather than any purity of soul. In Chapter One, for instance, Finny tells Gene, "You have a tendency to back away from things." And, later Gene himself narrates, "I suddenly became his collaborator."
So, Gene finds himself pulled out of his ingenuousness into "some equality for a while...." as they wrestle with each other and, then, compete with each other as Gene becomes sarcastic:
...this was my sarcastic summer. It was only long after that I recognized sarcasm as the protest of people who are weak.
From the summer on, Gene states that he
was subject to the dictates of my mind, which gave me the maneuverability of a straitjacket.
From this point on, Gene is consumed with envy and he loses his ingenuity as he rivals Finny. Interestingly, Ronald Weber writes in Studies for Short Fiction,
It is Phineas's innocence that Gene cannot endure. As long as he can believe Phineas shares his enmity, he can find relief; but, with this assurance gone, he stands condemned before himself and must strike out against his tormentor.
Indeed, it may be his own loss of ingenuousness taken from him by Finny as he perceives Finny's innocence that Gene envies the most.