For Irish Nationalists, the Irish state is composed of all 32 counties on the island of Ireland. The nationalist view is that 6 of these are currently occupied by the British (see Why don't Sinn Féin take their seats in the UK parliament?), while 26 are governed from Dublin. The legitimacy of the Dublin government is a point of contention among nationalists, in part due to the first Dáil of the Irish Free State "ceding" the 6 counties which form Northern Ireland to the British (the counties were already under British rule, as was the whole of Ireland). A view amongst republican Irish is that the third Revolutionary Dáil of the Irish Republic is the last legitimate Dáil as it was the last to be elected as a result of an "all-Ireland" election.
So to have a border control between the 6 counties in the North and the 26 in the South is as much a red line (for a nationalist) as if the French had demanded there be a border between Kent, Sussex and Surrey, and the rest of England, with those counties being part of France. No English Government would accept such a condition. The Nationalist point of view would see the Northern Irish border in the same way.
For Loyalists and Unionists in Northern Ireland the question is more pragmatic. Northern Ireland is a small region, and it is easier and cheaper to trade with Ireland, than to ship stuff over to and from the rest of the UK. For the economic development of "Ulster" they want the border as open as possible.
The open border is one thing that both Nationalists and Unionists agree on, for different reasons, and it is a key part of the Good Friday agreement.