While same-sex marriage was legalised in South Africa in 2006, the South African Anglican law on marriage states: "Holy matrimony is the lifelong and exclusive union between one man and one woman."
Bishop Raphael Hess, Tutu's senior in whose diocese her Cape Town base falls, said he was "vexed" by the need for her to renounce her clerical duties but that he hoped it would be short-lived.
"The time has come for us to exercise pastoral care, for us to demonstrate a shift that is reflected in the law," he said.
"We would be able to have Rev Canon Tutu be able to minister. At the moment she cannot and she has accepted that but we are hoping that there might be a window for us to change it."
Tutu is one of five children of Desmond Tutu, the former Archbishop of Cape Town who won the Nobel Peace Prize for standing up to the apartheid government and seeking to reconcile South Africans. He has also spoken out in favour of gay marriage, saying he would refuse to go to a "homophobic heaven".
Tutu is executive director of her parents' eponymous charitable foundation and divorced with two children. She married her long-time partner Marceline Van Furth, an atheist academic who is also divorced with children, in her native Netherlands in December.
The pair held a second ceremony that was attended by the Tutus and officiated by Rev Charlotte Bannister-Parker, a priest from Oxford, on Sir Richard Branson's wine farm in Franschhoek earlier this month.
Desmond Tutu was given permission by the Archbishop of Cape Town, the head of the Anglican church in Southern Africa, to give the union a "father's blessing".