Praise be to Allah.
Releasing hunting dogs against a gazelle that is tied up is a kind of torment and cruelty to animals, which is haraam. Al-Bukhaari (5513) and Muslim (1956) narrated that Hishaam ibn Zayd said: I entered with Anas upon al-Hakam ibn Ayyoob and there were some boys or young men who had taken a hen as a target and were shooting at it. Anas said: The Prophet (peace and blessings of Allaah be upon him) forbade taking animals as targets.
Al-Nawawi (may Allaah have mercy on him) said in Sharh Muslim: The scholars said: Taking animals as targets means tying them up when they are still alive in order to kill them by shooting arrows and so on. This is what is referred to in the words of the Prophet (peace and blessings of Allaah be upon him): “Do not take any living being as a target” i.e., do not take living animals as targets to shoot at. This prohibition means that it is haraam.
Hence the Prophet (peace and blessings of Allaah be upon him) said, according to a report narrated by Ibn ‘Umar: “Allaah has cursed the one who does that.” It is also cruelty to the animal, destruction of its life and a waste of its monetary value; it is a failure to slaughter it properly if it is an animal that can be slaughtered and eaten, and it is a waste of its usefulness if it is not an animal that can be slaughtered and eaten. End quote.
It was also narrated that it is forbidden to set animals against one another, but its isnaad is da’eef (weak).
It says in al-Mawsoo’ah al-Fiqhyyah (10/195): As for setting animals against one another, in the sense of setting one against the othe for the purpose of hunting, it is permissible with regard to trained dogs and similar animals. But there is no difference of opinion among the fuqaha’ with regard to it being haraam to set animals against one another and encourage them to fight, because this is foolishness and results in harm to the animal, and it may lead to it being killed for no legitimate purpose. It says in a report that the Messenger of Allaah (peace and blessings of Allaah be upon him) forbid setting animals against one another. End quote.
Al-Khaadimi said in Bareeqah Mahmoodiyyah (4/79): Examples of setting animals against one another include setting lions against tigers, or against cows and donkeys. End quote.
But if the hunting dog is released against the gazelle with the aim of training it and teaching it, there is nothing wrong with that, because it is a means to a permissible end.
And Allaah knows best.