In a video, aired on Aljazeera on Thursday, the group demanded that the UAE closes its embassy in Baghdad and that Iraq's Shia al-Fayha television channel halt operations.
Al-Fayha broadcasts from Dubai, in the UAE.
The group gave a 24-hour deadline to meet its demands.
The video showed Naji al-Noaimi standing against a bare wall, with his hands clasped in front of him.
Al-Noaimi was kidnapped by gunmen at about 10pm on Wednesday in Baghdad's al-Mansour district.
A Sudanese security guard, Bedawi Ahmed Ibrahim, who was shot by the attackers, later died of his wounds.
In July, an Egyptian and two Algerian diplomats were kidnapped and killed by al-Qaeda in Iraq.
In October, two Moroccan embassy staffers vanished, and al-Qaeda in Iraq said it had seized them and later sentenced them to death. There has been no word on their fate.