What is the Halacha in regard to whether or not a person can go hunting? What have Poskim said on this issue?
Revach explains the Teshuva referenced in the comments above:
When asked if hunting is permitted for sport the Noda B'Yehuda (YD 10) clearly expresses his distaste for this inappropriate question yet nevertheless answers the question on halachic grounds. He raises the possible issues of "Tzar Ba'alei Chaim" (causing pain to live creatures) and "Baal Tashchis" and rejects them both. Nevertheless he concludes that it is an extremely inappropriate activity for a Jew and there is an element of issur in it as well.
Hunting is not considered Tzar Ba'alei Chaim for two reasons. First, anything that provides any benefit for a human does not fall under the issur of Tzar Ba'alei Chaim. Second it is only assur if the animal is left alive in pain and not if it is killed.
With regard to Baal Tashchis it is only forbidden to destroy something that people would otherwise enjoy. Since these are wild animals and would continue to roam in the forest it does not constitute Baal Tashchis. Furthermore since the only possible benefit from wild animals are for their skins, killing them transforms them into something fit for human enjoyment.
While he says that one who truly hunts for his livelihood is certainly permitted to do so, no less that one who slaughters animals for meat and chicken, nevertheless, one who main purpose is entertainment should clearly refrain from such a cruel and inhumane activity. The only hunters we find in the Torah are Nimrod and Eisav not people we should be emulating.