In the Gospels, Martha was one of two sisters. She devoted herself to housework while her sister Mary sat and listened to Jesus. What is the irony here?

There are two sharp ironies here. The first is that while the Marthas are expected to do all the cooking and housework in Gilead, in the gospel story this refers back to, Jesus chided Martha for being too busy. He told her she should relax and come sit and learn from him, just as Mary was doing. Yet, in the novel, the Marthas are never encouraged to sit back and relax from their labors.
The other irony is that the kindness, compassion, and respect with which Jesus treated women, and in this particular story, Mary and Martha, is not extended to the women of Gilead. They are treated as property owned by men and as subordinate to men, placed on the Earth to service male needs. Yet, at the same time, Gilead portrays itself as a Christian state. We can only feel that Jesus would not approve.

The irony here is that the fundamentalist patriarchs in charge of Gilead, who claim to be such devout Christians, clearly don't understand Christ's message. Jesus himself said that Martha shouldn't trouble herself over such trivial things as housework; instead, she should take a leaf out of her sister Mary's book and serve Christ by listening to his Word.
Yet the elders of Gilead have determined that women—who, as far as they're concerned, exist purely to serve the interests of men—should preoccupy themselves with just such trivialities. Society's ruling patriarchs don't want women to get any ideas about interpreting the Bible for themselves, as they would soon discover that the life of mindless drudgery they're forced to endure has no Scriptural authority.
An additional irony here is that as devout Protestants, Gilead's patriarchs ought to believe in the freedom of every Christian to read the Bible for themselves. But they cannot and will not allow this to happen, as it would undermine their power and authority over women—just as the Catholic Church had its power and authority undermined by the Protestant Reformation, with its insistence on each and every believer discovering God's Word for themselves.

In the Bible, Martha is chastised by Jesus for worrying over things that are unimportant.  When Jesus comes, Martha chooses to try to prepare for him, busying herself about the house and fretting over all of the things that need to be done.  Her sister, Mary, however, chooses to sit and listen to Jesus's preaching instead.  Martha asks Jesus if he cares that her sister shirked her part of the household responsibilities, leaving her to handle them all alone, and he says that Mary had actually chosen the better way to spend her time.  This is ironic because, in The Handmaid's Tale, many women are assigned the duties of a "Martha."  These are the women who clean and cook and keep house for the Commanders and their wives.  Jesus said that this is not a good way for a woman to spend her time, that it would be better spent listening to his teachings, and despite the fact that Gilead claims to be founded upon biblical precedent concerning handmaids, it is clear that no one cares that Jesus admonished Martha.  It's ironic that the leaders of the community would adopt one such precedent and then ignore another.