Mr Burnham, the first practising Catholic to be Prime Minister, told Parliament that he would change the “antiquated” legislation.
The law removed bars on Roman Catholics and Jewish people holding public office but introduced a ban on people professing those faiths from “directly or indirectly” advising the monarch on appointments in the Church of England or Church of Scotland.
The Prime Minister announced his intention to change the law in a written statement to Parliament.
“It is unacceptable for there to be a legislative bar against people of any faith performing all the functions of my office,” he said.
“The Government will therefore introduce a Bill at the earliest possible opportunity to correct this antiquated and archaic anomaly.”
The legislation was set out for Roman Catholics in 1829 and for Jews in 1858, and anyone who breaches the rules would currently be “deemed guilty of a high misdemeanour” and banned from public roles.
It has prevented Mr Burnham from fulfilling a traditional role for prime ministers, forcing him to transfer responsibility to Alex Norris, the Lord Chancellor, to advise the King on appointments to roles such as the archbishops of Canterbury and York.
Mr Burnham has indicated various levels of commitment to his religious faith, saying in 2009 that it was alongside Everton FC and the Labour Party as one of the three important things in his life apart from family.
However, in 2015 he said: “I’m Catholic by upbringing, but I’m not particularly religious.”
A prime minister’s role in religious appointments is largely symbolic. According to official guidance from the Cabinet Office, approval from the Prime Minister and the King is only sought “once a candidate has been chosen [by a Church committee] and accepted the post”.
There have been instances when prime ministers have intervened, for example when Tony Blair, who became Catholic after leaving office, rejected both candidates for Bishop of Liverpool.
Gordon Brown, a Presbyterian, requested that the church submit only a single preferred name to be automatically sent to the monarch.
Sir Keir Starmer and Rishi Sunak were not barred from the role in their capacity as an atheist and Hindu respectively.
Boris Johnson, meanwhile, did not appear to delegate it despite being baptised as a Catholic. He was later confirmed as an Anglican but was considered to have reverted to Catholicism having had a Catholic wedding with Carrie Johnson.