An ambulance was called and the man, who was believed to have had an epileptic fit, was taken from the court room.
The three-month-old twins Chris and Cru Kahui died 12 hours apart on June 18, 2006, five days after Kahui and the twins' mother Macsyna took them to hospital.
Both had multiple injuries, including head injuries and brain damage, and one had a broken thigh bone.
The trial, at the High Court before Justice Geoffrey Venning, is scheduled to last seven weeks.
About 70 prosecution witnesses and 40 defence witnesses are expected to be called.
Police originally thought the twins died from "shaken baby" syndrome, but autopsies revealed their head injuries were the result of a blow or coming into contact with a solid object.
The twins were born prematurely and spent several weeks in hospital before being taken home to live with their parents.
But they were there only five weeks before being admitted to hospital and eventually dying.
