Question

What is the irony in the husband's refusal to lock the door

Answer 1

  Strong winds enter the house through the open door, and the husband tells the wife to "Gae [go] out and bar the door." 

She, in summary, tells him to forget it.  She's the one who is busy working, not him. 

They make a pact that whoever speaks first will be the one to have to close and bar the door.  They are both stubborn, and stubbornness is really what the poem is about.  You could say stubbornness is the theme of the poem.