What is the questioner's duty towards a gym where the Quran is played amidst noise and commotion, and is it permissible for him to continue training there?

The preferability of listening attentively to the recitation of the Holy Quran outside of prayer is the opinion of the majority of scholars. However, Quranic recordings should not be played in places of noise and idle talk, as this contradicts the veneration of the Book of Allah, and it is feared that it may be forbidden.

There is no blame on you for training in a gym where the Quran is being played, and you will not be held accountable for not listening attentively, because scholars who mandate listening have made an exception for places of preoccupation. The responsibility for playing Quranic recordings in such places falls upon the person in charge, and he should be advised and informed of the legal ruling.