Litvinenko, a Russian intelligence officer turned Kremlin critic, died in London in November 2006 after drinking tea spiked with the radioactive isotope polonium-210. His family says he was working for Britain's intelligence services, and believes the Russian state was behind his death.
Moscow authorities deny the claim, and refuse to extradite for trial two Russians identified by British authorities as the prime suspects in the killing.
Judge Robert Owen is due to oversee a coroner's inquest. Such inquests are held to determine the facts about violent or unexplained deaths.
Britain's government wants some evidence kept secret for national security reasons, a move opposed by Litvinenko's family and several media outlets.
A lawyer for Litvinenko's widow, Marina, complained Tuesday that the family and legal team do not even know what material the government wants to restrict.
"We are dancing in the dark," attorney Ben Emmerson said, accusing the British and Russian governments of conspiring to stop the truth from coming out.
Owen said Wednesday that he would examine that evidence behind closed doors, but promised to give the government request the "most stringent and critical examination." He said he could make the evidence public if he was not convinced of the government's case.
"It is my duty to carry out a full, fearless and independent investigation into the circumstances of the death of Mr. Litvinenko," the judge said. "That, I intend to do."
The inquest had been due to start May 1, but Owen conceded Tuesday that it would likely be postponed.