Burundi's rebels have ordered the most senior Catholic churchman in the country to leave within 30 days.
Monsignor Courtney had urged the rebel to give up their weapons
Archbishop Simon Ntamwana accused the National Liberation Forces (FNL) of killing the Vatican's ambassador to Burundi on Monday.
Michael Courtney, 58, from Ireland, was shot three times in an FNL stronghold, but the rebels deny responsibility.
Some 1,500 people attended a funeral service for him on Wednesday, before his body is flown back to Ireland.
"We ask the Catholic Church in Rome to find another host country for Simon Ntamwana in the coming days," FNL spokesman Pasteur Habimana told AFP news agency.
"We swear solemnly to the people of Burundi and the Church that we did not organise the ambush against the Vatican ambassador," he said.
He had played a key role in persuading other rebel groups to join the peace process and had urged the FNL to follow suit.
On Tuesday, Monsignor Courtney's sister had said that he had cancelled his usual post-Christmas break because he was optimistic about the peace process and wanted to help.
The Vatican has expressed "deep sorrow" at the diplomat's death, describing him as "a faithful and generous servant of the
Thabo Mbeki, president of South Africa, which has played a key role in the peace process, also expressed his "outrage and condemnation at the senseless murder of Monsignor Michael Courtney".
Burundi President Domitien Ndayizeye, speaking from the hospital where Monsignor Courtney died, vowed to bring those responsible to justice.
The Missionary Service News Agency (Misna) in Rome said that Monsignor Courtney had been travelling by car with three other people when their vehicle was sprayed with gunfire in Minago, Bururi province.
The other occupants escaped unharmed or with light injuries.
The Vatican expressed dismay at the monsignor's death
Mariano Benni of Misna told CNN that several sources had said the killing was an accident.
Monsignor Courtney had served as ambassador to Burundi since 2000.
Father Claudio Marano, a missionary in Burundi, told Vatican Radio that he had met all the various groups several times and the entire government.
"He was working a lot in favour of peace and reconciliation in Burundi," Father Marano said.
Some 300,000 people have been killed in Burundi's decade-old civil war, in which rebels of the majority Hutu
group are fighting to end dominance of the Tutsi minority.