The Bible doesn't specify that the killing of animals is morally wrong, per say. God seems to be the one who killed the first animal in the Genesis account to clothe Adam and Eve, not to mention all of the animal sacrifices that took place within the Old Testament, amongst other things. But I can greatly imagine what makes this action wrong when being done for sport today, is what manifests within the persons heart. I agree with a lot of others here - that being, killing an animal for the sake of food for themselves or for their family; for the sake of survival, would leave that person with a more grateful heart. Food after all, is a necessity to survival. However, I'm afraid that hunting for sport most time fills someone with pride, pleasure and seeking the thrill of the next kill. I can imagine it becoming a lust of the flesh, as the one who kills for sport and pleasure may never truly be satisfied no matter how big the game is. It becomes nothing more then a hunt for trophies at the end of the day. A senseless reason to kill anything.
With all that being said, I don't think that the killing of an animal would send someone to Hell, even if it was for sport. The killing of another human being is a different story, as we were the one's created in the image of God and God clearly says that murdering another person is morally wrong. God gave all human beings dominion over animals and I think He allows us to do with them as we please, whether that be wrong or right.
I think it's clear that God loves all of His creation and even the scriptures reference the animals waiting for the day that all things shall be restored because they too have been effected by this world of sin. I think that's a very beautiful picture to imagine, that the whole of creation will one day be in Glory together through the restoration of all things by the coming of the Lord Jesus.
"For the earnest expectation of the creature waiteth for the manifestation of the sons of God. For the creature was made subject to vanity, not willingly, but by reason of him who hath subjected the same in hope, Because the creature itself also shall be delivered from the bondage of corruption into the glorious liberty of the children of God. For we know that the whole creation groaneth and travaileth in pain together until now." Romans 6:19-22