Tough question. Finny is awesome. He's universally liked by just about everyone. He's good looking. He's athletic. He's funny and fun to be around. He genuinely wants to see his friends succeed. There aren't a lot of bad character traits about him. Finny isn't perfect, though.
Finny is a very spontaneous character. If something gets into his head, he goes for it. No planning or thoughts of consequences. While it might make him fun, it often gets him into trouble, too. He often gets caught doing things he shouldn't be doing.
Finny's biggest character flaw is tough to actually call a flaw. His fatal flaw is that he trusts everyone. He genuinely thinks that everyone else thinks the same way that he does. There isn't a jealous bone in his body. He wouldn't think about lying or cheating someone. The concept of competition is almost absent from his mind. He only sees the very best in people, and he thinks that everyone else is exactly the same way.
"Phineas was a poor deceiver, having had no practice"
So time and time again, the reader sees that Finny is genuinely looking out for what is best for Gene, despite the fact that Gene is usually looking out for what's best for Gene too. How to beat Finny is central to his thoughts. When Finny finally realizes that, it ends up inadvertently leading to his death.
It's odd that Finny's worst trait is that he's just too good.