In Fahrenheit 451, Beatty says "[Fire] is a mystery...Its real beauty is that it destroys responsibility and consquences." How has this been true?
Beatty says this when he and Montag are at Montag's house, preparing to burn it down. In the earlier parts of the book, we can see how fire (or at least the way fire has been used in this society) has destroyed responsibility and consequences.
In this society, fire has been used to burn books. This is part of a larger drive to rob people of their human impulses. People have been encouraged to ignore the real problems that come with being human. They have been encouraged to simply rid themselves of these problems by pretending they don't exist. True human life consists of having feelings and problems and confronting those things and their consequences. In this society, by contrast, people run from those things. We can see this clearly in how Mildred retreats into her parlor walls and in how her friends try to forget their families.
When Beatty says the line you cite, he is speaking figuratively. His society's default reaction to every problem is to burn it (literally or figuratively). He is saying that they rid themselves of responsibility and consequences by simply destroying their need to make choices and confront problems. This society does this by trying to get people to simply live in the moment and forget that they are human beings.