Ben Shaw, 15, was attempting to bench press almost 100kg (220 lb) when the accident happened in Brisbane on Tuesday, local media said.
He was found by staff and taken to hospital, where he died on Saturday. It is not known how long he was trapped.
"Ben passed away yesterday afternoon with his family and friends around him," his family said in a statement.
"Ben was able to leave a legacy and donate his organs and tissue, giving life to others."
Authorities said it was too early to say whether charges would be laid, but they would conduct a thorough investigation.
According to rules at the Police Citizens Youth Club gym, people under 16 must be supervised when lifting weights.