There are concerns in the party it could lead to an early election for the Northern Ireland Assembly.
Removing Arlene Foster is a “total mess” that could lead to a Northern Ireland Assembly election this year, a senior DUP source has told the BBC.
Mrs Foster will step down as Democratic Unionist Party leader on 28 May and as first minister at the end of June.
She had faced an revolt with about 80% of the party’s Stormont and Westminster ranks backing a change of leadership.
The next Assembly election is not due to take place until May next year.
A new DUP leader is likely to be elected before the end of May but a senior source says Sinn Féin may demand concessions on the Irish language or LGBT rights before agreeing to the election of a new first minister.
“I don’t think any of them have gamed out what is likely to happen now,” said the source, who asked not to be named.
Sinn Féin’s Michelle O’Neill – the joint head of Northern Ireland’s government along with Mrs Foster – said the “incoming DUP leader should recognise that the political landscape across our island has changed”.
“Within the executive and assembly, Sinn Féin will work with all parties to progress social reform, political change and economic prosperity – but we will robustly oppose damaging policies or regressive throwback politics of the past.”
The DUP source said the words were a warning to his party.
Without Sinn Féin support a new first minister cannot be elected.
The DUP source told the BBC: “I would expect negotiations to take place over the election of a new first minister but Sinn Féin will seek to extract movement on a number of issues.
“Unless there is a agreement I can see the issue being allowed to drift over the summer when not much happens and then a possible election in the autumn.”
There has been speculation of a split role, with one person for first minister and then a second person as party leader at Westminster.
Possible contenders include NI Agriculture Minister Edwin Poots, who has been tipped to take over as first minister, while Lagan Valley MP Sir Jeffrey Donaldson and East Belfast MP Gavin Robinson could vie for the party leader role.
Mr Poots tweeted that he wanted to thank Mrs Foster for her “service, sacrifice and commitment to the DUP and country”.
Mr Robinson thanked Mrs Foster for her “dedicated service to Northern Ireland” and said there would be “more to say about what lies ahead in the days to come”.
Only a small number of the DUP membership – MLAs (Members of the Legislative Assembly) and MPs – will get to vote in a leadership contest.
There are 27 MLAs that hold the party whip – including Mrs Foster – and eight MPs.
If a contest does take place, it will be the first in the DUP’s 50-year history.
DUP rules say a leadership election should be called by 30 April, although that date could now potentially pass by.
Party officers will ultimately decide the selection process.
Normally the leadership election is simply a formality and the existing leader continues in their role, but other candidates are free to challenge them.
The role of first minister is separate, but as the largest party from the largest grouping (unionism) in the assembly, the DUP is entitled to nominate someone to hold the post of first minister.
Once Mrs Foster steps down as first minister at the end of June, the DUP would then be asked to nominate a replacement within seven days and a vote would be held in the assembly.
If a replacement is not nominated, then the Northern Ireland secretary is obliged to call an assembly election.
Mrs Foster has endured a turbulent time as DUP leader and the fall-out from Brexit – which the party supported – has put particular pressure on the party’s top brass as it faces having to weather the storm caused by the Northern Ireland Protocol, which imposed a border down the Irish Sea.
The aftermath of Brexit caused friction both internally and between the party and some of its core voters who are unhappy at the deal which led to the Irish Sea border.
It has also been suggested recent changes to NI’s abortion laws and the commitment to implement an Irish language act were causing concerns with some elected DUP representatives ahead of next year’s assembly election.
On Tuesday it emerged a letter of no-confidence was circulated among the party’s MLAs and MPs. More than 20 assembly members and four MPs are understood to have signed it.
There was also pressure from councillors, with concerns ranging from the NI Protocol, abortion legislation, same-sex marriage, policing and the recent gay conversion debate at Stormont.
Mrs Foster, 50, had been DUP leader since December 2015.
The following month, she was appointed first minister of Northern Ireland, becoming the first woman and the youngest person to hold both jobs.
Announcing her resignation on Wednesday, Mrs Foster said it had been the privilege of her life to “serve the people of Northern Ireland as their first minister and to represent my home constituency of Fermanagh/South Tyrone”.
Colleagues from across the political divide have extended their best wishes to Mrs Foster.
Deputy First Minister Ms O’Neill said the pair had a decent, if challenging relationship.
It is understood Mrs Foster and Ms O’Neill will still jointly chair a meeting of the Stormont executive later on Thursday.
When the two leaders spoke on Wednesday they said it was important the meeting happen, ahead of Covid restrictions being further eased in Northern Ireland on Friday.
NI Secretary Brandon Lewis said Mrs Foster was a dedicated public servant who had inspired young women to follow her example and go into politics.
Taoiseach (Irish PM) Michéal Martin said political leadership was often not easy and took courage.
“I have had a positive working relationship with Arlene in a variety of roles over the years,” Mr Martin said.
“In particular I valued the constructive engagement, notwithstanding our differences, we have had in our respective roles as taoiseach and first minister.”