"Will remember you as the cheeky funny but clever boy in the class at Manukorihi and High School," Mel Waters wrote in memory of Mr White.
Joseph White wrote on Facebook: "Words cannot express the pain nor emptiness that is engulfing all our hearts and minds at the moment. He was a true gentlemen, a valiant young man taken too early."
Another friend said: "Cannot express how much sadness we feel - we love you brother Ben and our hearts are with you Liz and babies."
Mr White, according to his Facebook page, is survived by his wife Elizabeth and their young children. He grew up in Waitara and attended New Plymouth Boys' High School.
Kwinana Refinery is on the shore of Cockburn Sound at Kwinana, near Fremantle, Western Australia.
Conflicting reports suggest Mr White fell between 10m and 40m.
The Department of Mines and Petroleum, which is investigating the incident, said Mr White entered a digester vessel that had no scaffolding, WA Today reported.
He fell a "considerable distance" to the bottom of the vessel, it said.
Mr White was a contractor at the refinery.
The company said no other employees or contractors were injured in the incident and Alcoa was working closely with the relevant authorities.