Lord Janner had been Board president from 1978 to 1984. A Board spokesman said: "Following the passing of Greville Janner after a long illness, our thoughts and prayers go out to the whole Janner family."
Greville Janner QC, Lord Janner of Braunstone, held the Leicester West seat for Labour for 27 years before stepping down in 1997. The father-of-three was a founding patron and chair of the Holocaust Education Trust and served as a vice-president of the World Jewish Congress. He held a number of leading positions on groups serving world Jewry.
Karen Pollock, Holocaust Educational Trust chief executive, said: "In 1988 Greville Janner had the foresight to know we as a nation needed to know about and remember the Holocaust and so established the Holocaust Educational Trust. Our thoughts are with his family at this difficult time."
Lord Mendelsohn, Commonwealth Jewish Council president, said: "Greville's contribution to Commonwealth Jewry will be long remembered. Our thoughts and prayers are with the family."
Yiftah Curiel, spokesman for the Israeli embassy in London, said the embassy was "saddened to hear" the news of Lord Janner's death.
Mourners gathered to attend Lord Janner's funeral on Monday afternoon.
He was laid to rest at a United Synagogue cemetery in Willesden, north-west London, in front of family, friends and colleagues from the community. One mourner estimated that there were around 300 people in attendance.
Fellow veterans from the Association of Jewish Ex-Servicemen and Women (Ajex) also attended. Lord Janner served in the army and later acted as president of the association.
An ongoing legal case had been investigating claims against the peer relating to historic child abuse, but a series of judges had ruled he was not fit to stand trial, most recently just a fortnight before his death.
A "trial of the facts" was due to take place in April 2016, which would allow a jury to decide whether he had committed the offences, but with no finding of guilt or a conviction.
A hearing will take place in the High Court, probably in mid-January, to decide what happens next. The Crown Prosecution Service said it was "considering the procedural implications" following Lord Janner's death. It would be unprecedented to proceed with a case after a defendant had died.
The peer had faced 22 charges relating to a series of sexual offences, the majority against young boys under the age of 16, in the 1960s, 1970s and 1980s.
The allegations included 15 counts of indecent assault and seven counts of a separate sexual offence, against a total of nine complainants. It was reported this week that three further complainants had come forward and more charges were being considered.
Liz Dux, a specialist abuse lawyer representing a number of the alleged victims, said her clients felt "deeply frustrated. All they have ever wanted is to give their evidence in court".
The case is expected to be included in the wide-ranging Goddard Inquiry into child sexual abuse. Lord Janner's accusers may also take civil action against his estate for compensation.
Lord Janner's family had repeatedly maintained his innocence.
Born in 1928 in Cardiff, Greville Janner was the son of MP and Anglo-Jewish leader Barnett Janner.
He attended St Paul's School in London, and Trinity Hall, Cambridge, becoming president of the Cambridge Union, and later winning a Fulbright scholarship to study at Harvard Law School.
He became a barrister in 1954. As a Labour MP, he followed his father in representing Leicester North West, winning the seat in 1970. He held it in 1974 when it was renamed Leicester West.
