You don't specify whether you need Bible verses that actually appear in the novel, or whether you need Bible verses that are applicable to the novel. I am going to assume the latter, as it seems to make more sense.
If you think about the themes and motifs in this novel, you will be able to find verses in both the Old and New Testament that relate to it. For example, there is the issue of adultery, at the center of the novel. If you do an on-line search for "adultery" + "Bible verses" you will come up with many. For example, from the Old Testament:
(Exodus 20:14) 14 “You shall not commit adultery.
And, from the New Testament:
(Matthew 5:27) “You have heard that it was said, ‘You shall not commit adultery;’
Now, think about some other ideas that are brought out in this novel. If you look them up as well, you will find verses that relate to these themes/ideas in both the Old and New Testaments. Below is a helpful link that arranges the verses by topic. Some that you might check out, that apply to this novel are: vengeance, forgiveness, healing, love, hate, evil and especially sin.
Of course the Old Testament's "an eye for an eye" (Exodus 21: 23-25) immediately comes to mind, but so do verses from the New Testament as being appropriate to "The Scarlet Letter's" Roger Chillingworth's violation of the human heart as he seeks revenge against the Reverend Arthur Dimmesdale:
Judge not, that ye be not judged/For with what judgment ye judge: and with what measure ye mete, it shall be measured to you again./And why beholdest thou the mote that is in thy brother's eyes, but consider not the beam that is in thine own eye?/ (Matthew 7:1-4)
Of course these lines can also apply to the Puritans who judge Hester so severely but have secret sin in their hearts. One example is the Governor, whose sister is known to go into the forest for the Black Mass. The following lines from Matthew also apply:
Thou hypocrite! First cast out the beam out of thine own eye and then thou shalt see clearly to cast out the mote out of thy brother's eye. (Matthew 7: 5-6)
Certainly, the humbled Hester learns to follow the "golden rule" of Matthew 7:
Therefore all that you wish men to do to you even so do you also them (13-15)
And, the townspeople learn to revere Hester, seeing the once disgraceful letter as meaning "Angel":
Wherefore by their fruits ye shall know them (Matthew 7: 20)
In the final scaffold scene, Chillingworth is unable to harm or reach Dimmesdale any more. When the malevolent physician says, "Thou hast escaped me," these lines come to mind:
...depart from me, ye that work iniquity (Matthew 7:23)