What are two internal conflicts that Gene has that he covers up with a lie that results in fear and guilt?
I would say the larger of the two internal conflicts that Gene has and has to cover up with a lie is the conflict about whether or not he bounced the branch and knocked Finny off it. Yes, Gene did bounce the branch, and in a round about way that single event is what ended up killing Finny. Throughout the novel, though, Gene is constantly trying to convince himself and others that he did not bounce the branch intentionally or unintentionally. Gene is terrified that other students will figure it out.
The branch bouncing incident makes Gene feel guilty in two specific ways. First, he feels guilty about having done it and caused Finny so much harm. It was way worse than a simple fall. Second, he feels guilty because Gene is learning that Finny truly is as altruistic as he first appears. Gene just can't understand how Finny doesn't harbor a jealous bone in his entire body. That makes Gene feel worse for his selfish branch bouncing.
A second internal conflict is what to tell the others about Leper. Late in the novel, Gene visits his friend Leper, who has come home from the war. Leper is AWOL and no longer completely sane. Gene knocks Leper out of a chair when Leper accuses Gene of having knocked Finny out of the tree. Gene feels guilty about abusing a mentally sick friend of his, especially knowing that Leper is right. Part of Gene fears that Leper might tell other people his thoughts about Gene.
Gene returns to campus and Finny and the other boys ask about Leper. Gene is forced to gloss over the details and hide some of the specifics. It's not exactly an overt lie. Gene didn't say anything false, but he deliberately hid facts from the other boys to protect himself. The other boys are smart and figure out that Leper has gone crazy, but they don't know about the entire exchange.