The fire was set by police and military trying to remove a group of indigenous farmers, peasants and university students who took over the mission to denounce repression by the armed forces during Guatemala's civil war, which began in 1960 and ended in 1996.
Pedro Garcia Arredondo, the sole defendant, is accused of murder, attempted murder and crimes against humanity.
Some 200 people, many of them victims' relatives, turned out for the Guatemala City trial, including Menchú, the 1992 Nobel laureate.
"We hope that the courts operate under the law and that the judges are not pressured," Menchú told AFP.
"We want to finally close the cycle of pain, of our suffering. This load is painful," she said.
An Indian priestess and Menchú carried out a Mayan ceremony in front of the court before the trial, calling for a good start to the proceedings and for justice to be done after more than 34 years.
The Spanish ambassador to Guatemala, Manuel Lejarreta, told AFP that his government was not a plaintiff in the case but "is interested in how it goes."
He said he was confident that the Guatemalan courts can get to the bottom of the burning of the embassy, which killed four Spanish diplomats including the consul.
Around 200,000 people died during the Guatemalan civil war, according to the UN.