Why did Gene tell Finny the truth about pushing him off of the tree?

Gene had to. Ironically, Finny always really knew the truth. The truth was always swept under the carpet, so to speak, and replaced with cryptic suggestions about what might have happened. Gene's guilt was obviously too much. More than that, Gene felt an open confession, without any excuses, was needed for some kind of cleansing. As long as a full confession was not made, there was a certain amount of toxicity in their relationship. It prevented the boys from becoming each other (I know that sounds strange but it is one of the themes of this novel.)