The party went from 37 DÃ¡il (Irish parliament) seats in 2011 to just seven.
Ms Burton was TÃ¡naiste (Deputy Prime Minister) and Minister for Social Protection in the last government.
She took over from Eamon Gilmore, two years ago, but will stay on until her successor is chosen.
The front runner is Brendan Howlin, from Wexford, who is the former Minister for Public Expenditure.
But it is understood that he only wants the job of Labour leader if he is the unanimous choice.
Other possible candidates include Alan Kelly, the former environment minister, and Sean Sherlock, a junior minister at the Department of Foreign Affairs who was involved in the Fresh Start Agreement talks in Northern Ireland.
If there is a contest it could be late August before the outcome is known.

Dominic Colella was called out on 9 March 2013 when the 85-year-old man collapsed in a queue at a London branch of the store.
A colleague told a disciplinary hearing Mr Colella had returned to the ambulance with two "full bags of shopping" after 20 minutes.
He did not attend the hearing.
Giving evidence to the Health and Care Professionals Council, fellow paramedic Yvonne Purves said the elderly man had passed out in M&S and then regained consciousness before being moved into Mr Colella's ambulance.
She told the panel she had wondered where her colleague had gone after waiting for some time in the ambulance.
"The wife was understandably anxious about what had happened to her husband and was wondering why we weren't leaving to go to hospital," she said.
"He then came back with two full bags of shopping and loaded them in to the front of the ambulance and flippantly said 'do you want to go to hospital then?'.
"Obviously, I wanted to go to hospital some time ago. I waited 20 minutes for Mr Colella while alone in the ambulance," she added.
On another occasion, Mr Colella is alleged to have got a hair cut instead of taking a head injury patient to hospital.
The panel heard Mr Colella, who faces two counts of misconduct, did not turn up at the hearing because he was working.
He has since resigned from the London Ambulance Service.
The hearing continues.

The five were a 16-year-old girl, who died in Great Victoria Street on Monday, and four men who died in different parts of the city.
Police did not confirm which three cases are suspected to be drugs-related but no single, common drug is involved.
Earlier, a senior officer said all five deaths may have involved drugs.
But a Police Service of Northern Ireland (PSNI) spokeswoman later clarified that detectives suspect drug misuse may have been a factor in only three of the deaths.
Post-mortem examinations are being carried out and toxicology reports have been ordered.
The first of the five sudden deaths took place in Verner Street in Belfast city centre on Friday, 31 March.
The other four people, including the teenaged girl, all died on Monday:
Police said that number of deaths in such a short time frame was unusual, but that they were not linking any of them at present.
Ch Supt Tim Mairs said: "Investigations are at an early stage, but I would like to take the opportunity to reinforce the dangers posed by taking any illegal drugs or any medication that is not prescribed for you.
"Drugs misuse can affect anyone and we recognise that many vulnerable people are being exploited for the financial gain of those higher up the supply chain."
He added: "The number of deaths in Northern Ireland connected to drug use is on the increase and this is a concern.
"Sadly, these are all preventable. In fact, more people are dying from misuse of opioid drugs in Northern Ireland than in road fatalities, which is a very stark reality."
The leader of the Irish Labour party, Joan Burton, is to stand down after the party's poor performance in February's general election.
[NEXT_CONCEPT]
A paramedic shopped in Marks and Spencer while an elderly patient with severe blood poisoning waited in an ambulance, a conduct panel has heard.
[NEXT_CONCEPT]
Five sudden deaths in greater Belfast over the past few days are being investigated, with police saying three of the cases may have involved drugs.