Foster, 50, grew up during the decades of Northern Ireland violence known as "the Troubles", and as a child saw her father, a part-time police officer, shot and wounded in an Irish Republican Army attack.
The DUP is facing anger from the wider loyalist and unionist community over the rollout of the Northern Ireland Protocol which critics argue has created an economic and trade border in the Irish Sea between Northern Ireland and the rest of the UK.
Critics have accused Mrs Foster of failing to use the party's influence in Westminster - particularly during its confidence and supply deal with the Conservatives - to secure a Brexit deal that saw Northern Ireland leave the European Union on the same terms as the rest of the UK.More news: CDC may soon update outdoor face mask wearing guidance
On Tuesday evening it was reported that the letter stated that the signatories no longer had confidence in "Mrs Foster or the leadership", which has been interpreted as an attempt to also remove Lord Dodds as the party's deputy.
Foster also has alienated sections of the conservative, Protestant party by taking a too-liberal stand on social issues.
"I want to thank Arlene Foster for her dedication to the people of Northern Ireland over many years", Johnson said in a tweet. The Belfast administration collapsed in January 2017 over a botched green energy project.
Northern Ireland's government resumed work at the start of 2020, but deep distrust remained on both sides.More news: United Kingdom defense secretary denies ‘deeply unfair’ claims on Boris Johnson
This leadership election will be held over the coming weeks.
Foster was the first woman to lead the DUP, a party rooted in the fundamentalist Free Presbyterian Church.
The announcement comes a day after a majority of DUP lawmakers signed a letter seeking a confidence vote in Foster amid discontent over her leadership, including her perceived mishandling of Brexit negotiations.
I understand the misogynistic criticisms that female public figures have to take and sadly it's the same for all women in public life. The party also wants the United Kingdom government to scrap its divorce agreement with the EU. It is unclear how many have signed, and the DUP said it would not comment on internal party matters.More news: Australian city of Perth to enter snap 3-day lockdown