Cardinal Burke on irregular marriages.

The Church has always allowed that.

In an interview on "Church Chat" Cardinal Burke stated:

“In the whole history of the Church, it’s never been possible that someone who was living publicly in a state of sin, for example a person who is bound in matrimony to one person, is living in a marital way with another person, it has never been permitted that such a person could approach to receive Holy Communion,” the cardinal said in his latest interview. 

“Now, suddenly, there are those, who are saying, ‘Oh, yes, but it is possible in certain cases,’ and so forth - but if this is something that is always and everywhere wrong, how is it possible that someone who is living this way can receive the Sacraments?” 

Burke said couples in such situations - for a just reason - may receive the sacraments if they refrain from sexual situations.

“The Church has always helped such couples,” he said, “… the point is this: that the couple has the firm resolve to live chastely, and to take all measures to live chastely. If they fail, on one occasion or another, then they simply have to confess that, and renew their effort to live chastely.” - Source

A man commenting online about the situation Burke referred to, said that is like giving a kid an ice cream cone and telling him not to lick it.

That's not true.  People have done that, people do that, and people will continue to do so - with the help of God's grace.  Quite seriously, their love deepens - to love another is to desire what is best, what will make them happy, and to do all you can to do that.  Love of God and one's eternal salvation is the greatest good, to receive Christ in the Blessed Sacrament is the greatest happiness on earth.  I know people who have respected one another so deeply, as to willingly consent to live in such continence.  Neither God, nor the Church and her ministers can condemn such devotion.

Disinterested, but loving.