The ALP failed to capitalise on the Liberals' 12 per cent primary vote collapse, gaining about one per cent in Saturday's poll.
Ms White on Sunday conceded her party would not be able to form government, marking a third loss in her attempt to become premier.
"I'm very proud of the campaign that we ran this election but obviously we didn't make it and I take responsibility for that," she told reporters on Tuesday.
"It's time for me to move on.
"It's time for change and time for renewal of the leadership of the Labor Party."
Dean Winter is one of the favourites to replace Rebecca White as Labor leader. (Ethan James/AAP PHOTOS)
Frontbench MPs Dean Winter and Josh Willie are the leading contenders to lead the party.
Ms White will remain in parliament and support her successor, including in the shadow ministry if asked, who will have the "great opportunity" to hold the government to account.
"There's no doubt they've lost a lot in this election campaign, not just in terms of support across the electorate but credibility too," she said of the Liberals.
"The Labor Party now has an opportunity to take it up to them and they're in a good position to win the next election."
Mr Winter said Ms White was a "relentless, tough and intelligent leader" who never stopped working or gave up.
"The thing that sets her apart from anyone I've ever worked with is her work ethic," he said.
A mother of two, Ms White was first elected to the rural electorate of Lyons in 2010 having attained a commerce and arts degree.
Rebecca White has been a tireless and powerful champion of working Tasmanians.— Julie Collins (@JulieCollinsMP) I know Rebecca will continue her fierce advocacy of her constituents and hold any future government to account on behalf of all Tasmanians.I thank her for her leadership and service to Labor values. pic.twitter.com/6A0TbRuRp7March 26, 2024
After former leader Bryan Green resigned, she took on the party's leadership in 2017.
Ms White stepped down after Labor's 2021 defeat but was reinstated three weeks later when David O'Byrne quit over historic sexual harassment complaints.
Mr O'Byrne, who remained in parliament as an independent, is set to be re-elected and could be in a position to prop up a Liberal minority government.
Federal Tasmanian MP Julie Collins described Ms White as a "tireless and powerful champion of working Tasmanians".
While she has not publicly backed any candidates for leader, Ms White praised her deputy Anita Dow, who was the first Labor woman elected to the seat of Braddon.
"She has achieved many firsts and broken through many glass ceilings and done it quietly," she said.
"She has my full support whatever she chooses to do next."
Ms Dow thanked Ms White for her "enormous contribution", calling her "one of the most intelligent, hard-working, tenacious and caring people".
Ms White has praised her deputy Anita Dow. (Rob Blakers/AAP PHOTOS)
Counting continues but with both major parties short of the 18 seats needed for majority government, the result means a hung parliament.
Jeremy Rockliff, who is expected to remain the Liberal Party's only premier nationwide, has reached out to the Jacqui Lambie Network, which secured two and possibly three seats, as well as two independents, in order to assemble another minority government.
As it stands, the Liberals have 13 seats, Labor 10 and the Greens four, with four doubtful.
Ms White remains adamant Labor, under new leadership, will not do any deals with the minor parties or independents to wrest power.
Tasmania went to the polls more than 12 months early, with Mr Rockliff seeking a return to "stability and certainty" after spending the best part of a year in minority.
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