However, it should be a burning issue for every other party as well - because voters are entitled to know what sort of coalition they might get after the next general election.
At Labour’s first election hustings on Monday night, Burton (below) made her position clear. As far as the frontrunner is concerned, there is still a “Sinn Fein-IRA nexus” that puts Sinn Fein beyond the pale.
Her would-be deputy Alan Kelly put it even more bluntly by slamming Gerry Adams’s party as “the enemy”.
Alex White has taken a much more convoluted stance. On the one hand he says it would “stick in my craw” to do business with Sinn Fein, on the other he argues that they cannot be treated as “untouchables”.
He has also claimed that forming a government is all about Dail arithmetic and nobody can afford to rule the Shinners out.
Just a few weeks ago, this might have seemed like an academic debate.
But everything has been changed by last month’s elections, in which Sinn Fein trounced Labour by hoovering up the left-wing vote.
They have already secured the Lord Mayor of Dublin’s job for 2016 - and now they have their hearts set on grabbing the Taoiseach’s office as well.
Anybody who thinks Sinn Fein’s result was a flash in the pan is sadly deluded. The first post-election opinion poll has shown them rising to 26pc, more than Fine Gael (20pc) and Labour (5pc) combined.
If - and it is a huge if’ - those figures were repeated in a general election, it would be virtually impossible to keep the Shinners out of power.
So who might do business with them?
Enda Kenny has consistently ruled Sinn Fein out as coalition partners, although he struggled to explain why in a disastrous Late Late Show interview back in 2010.
Micheal Martin regularly accuses them of glorifying IRA violence and having crazy economic policies, but some more republican members of his frontbench would happily serve in a Fianna Fail-Sinn Fein partnership.
All these people should proceed with great caution. Sinn Fein may be on a roll these days, but they are still the Marmite of Irish politics with most people either loving or hating them.
Sharing power with the Shinners would be like handling a lighted firework - something very difficult to do without getting your fingers burned.
No amount of remarks from high minded apologists who defend or try to rationalise the murderous actions of the IRA, can disguise the fact that Sinn Fein is still haunted by its bloody past.
Gerry Adams was recently arrested for questioning over a 42-year-old murder, the sort of scandal that would destroy any other leader.
At Westminster this week, the party is refusing to give evidence about its so-called on-the-runs’ - IRA fugitives who want an amnesty for any crimes they may have committed.
This is why Joan Burton’s declaration that she will not touch Sinn Fein with a barge pole looks like a smart move.
It marks her out as a woman of principle, something that Labour could have done with a lot more of over the last three years.
If Burton becomes the next Tanaiste, she should consider asking Enda Kenny and Micheal Martin to sign a joint pledge against their common enemy.
By taking such a clear-cut stand, they could show that forming a coalition is not just about maths - it has to have a moral dimension as well.
Alex White is way behind in the Labour leadership race. To stand any chance of overtaking Joan Burton, he obviously needs to take some risks.
Holding the door of Government Buildings open for Sinn Fein, however, may well be a risk too far.