Burke, then aged 18, used a kitchen knife to stab Abraham Baez Rodriguez's neck, severing a major artery and causing him to bleed to death, prosecutors said.
Fellow Dubliner Stuart beat Laura Negrin Garcia round the head with a large stone, knocking her unconscious, it was claimed.
State prosecutor Angel Garcia Rodriguez told the jury: "What started out as a simple argument ended with Abraham Baez dead with a knife in his neck and Laura Negrin injured by a stone."
The prosecutor said Burke knifed 25-year-old Abraham "with the intention of ending his life", and said the Irishman should be jailed for 17 years for murder.
The Irish couple, who have since split up, went on trial yesterday at the Provincial Court in Santa Cruz, the island's capital. They travelled to Tenerife together from Ireland and deny any wrongdoing.
Burke has been on remand since the tragedy. State prosecutors originally charged Stuart with murder but dropped that charge. They now want her jailed for three years for assaulting Laura.
Burke, in handcuffs, wept during parts of the hearing, as did Stuart, who wore a white blouse and black trouser suit, with her hair in a bun. They were returning to their holiday apartment after a night out in the party resort of Playa de las Americas when the fight broke out at 6am on April 10, 2010.
Burke had downed at least 10 drinks including beer and vodka, while his girlfriend had drunk a bottle of wine, 10 vodkas, beer and shots.
The Spanish couple, who had been on a night out with friends, had drunk at least 10 whiskies each, the court heard. The fight broke out outside the Hotel Tropical Playa when Laura opened a car door, accidentally banging it into Sara, the jury was told.
Prosecutors say there was a fist-fight before Burke went off to the couple's apartment, returning with a wooden-handled kitchen knife with a four inch blade. Abraham collapsed in a pool of blood after being stabbed in the left side of his neck. The blow severed his carotid artery, and paramedics were unable to revive him.
His girlfriend Laura told the court she was knocked out by a blow to the head with a blunt weapon. She added: "When I came around, Abraham was lying on the floor."
Burke denied stabbing Abraham and told the court he had been attacked by the Spaniard.
He claimed police had "tortured" him and beat him with batons until he confessed to attacking Abraham.
Sara (31) also claimed she had been attacked and denied striking Laura with a stone.
She told the jury: "The girl opened the car door and knocked me onto the ground.
"She got out of the car and we fought. She hit me and I hit her back. But I didn't hit her with a stone." The trial continues.