All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
In principle, it is forbidden to make an animal or bird a shooting target. Ibn 'Abbas narrated that the Prophet said: “Do not make a living creature as a target.” [Muslim] Ibn 'Umar said, “that the Prophet cursed anyone who makes a living thing into a target.” [Muslim]
As for your question, we say that a pigeon is not a wild animal in most cases. So, it is not permitted to eat it unless it is slaughtered according to Islamic Sharee’ah, i.e. by cutting its throat and jugular veins. In other words, although it is forbidden to hold such a competition or game of shooting live animals, when the game comes to an end and you find that the animal is still alive, it is lawful to slaughter it Islamically, and then to sell it. However, if it is found dead, then it becomes forbidden to benefit from it even if it is killed by hunting, since it is not slaughtered Islamically as being a tamed animal. In addition, we are to know that only untamed animals become lawful by hunting. A fleeing tamed animal that we can not slaughter Islamically also becomes lawful to hunt.
As a final word, it becomes clear that slaughtering pigeons that are found dead in the above-mentioned games as well as selling them to Muslims constitutes a big deception and treason.
Allah knows best.