In Chapter XIII, Hawthorne writes as a separate paragraph,
The scarlet letter had not done its office.
In this chapter, Hawthorne describes how Hester has changed: The warmth, and passion appear to have been replaced by distance, severity, and grayness; her hair has lost its luxuriant color while her beauty, too, has faded. Similarly, she retreats from company, isolating herself, and she acts more like a servant than a nurse when she cares for others.
Moreover, the repentance that the church leaders have expected to exact from her is instead a cold despair. In fact, Hester considers sending Pearl
at once to heaven, and go herself to such futurity as Eternal Justice should provide.
By the fact that Hester considers suicide, which is an unpardonable sin in the Puritan code, Hawthorne indicates that "the scarlet letter has not done its office" and she has been untrue to her letter.
The previous answer is exactly right. One other time in The Scarlet Letter when Hester is untrue to the token of shame is when Pearl asks her mother why she wears the scarlet letter and she lies. Pearl, as you know, is a rather precocious child who has the uncanny ability to prick her mother's conscience at every turn, deliberately or not. She does such things as throw cockle burrs at her mother's bosom, and she is flighty and rather disobedient at every turn. When Hester lies and says she wears the letter because it's beautiful, she is, indeed, untrue to the letter; but we understand why. The real reason would have given Pearl even more ways to hurt her mother, even though she may not mean to do so.
The only part of the story in which I can think of Hester being untrue to the scarlet letter is towards the end of the story, when she and Dimmesdale get together in the forest and plan to escape together out of the country with their daughter, Pearl. At this point,a sense of peace and joy enters Hester for the first time since her punishment of wearing forever the big scarlet letter. When she felt this way, she took off her bonnet, let her hair down, went with the flow, and took off the letter as well. This letter had bound her to a life of misery and, since she isn't miserable in that specific moment, she took it off in a symbol of freedom from it. However,we know that Pearl did not sit well with this, and could not recognize her mother without the letter. For this reason, Hester had to put the letter back on.