It depends on where the lie has been said and in which circumstances.
For example, Karna lied to his Guru Parashurama about his caste. Now, such a lie is considered by Manu as a grave offence.
11.55. Falsely attributing to oneself high birth, giving information to the king (regarding a crime), and falsely accusing
one’s teacher, (are offences) equal to slaying a Brahmana.
So, in this case, Prayaschittas that are prescribed for slaying a Brahmin, has to be performed.
But, in normal circumstances it is an Upapataka. And, since it involves an impurity of the speech, the Goddess of speech or Goddess Saraswati has to propitiated as an atonement.
See the following verses:
8.104. Whenever the death of a Sudra, of a Vaisya, of a Kshatriya, or of a Brahmana would be (caused) by a declaration of the truth, a
falsehood may be spoken; for such (falsehood) is preferable to the
8.105. Such (witnesses) must offer to Sarasvati oblations of boiled rice (karu) which are sacred to the goddess of speech, (thus)
performing the best penance in order to expiate the guilt of that
105 is talking about an atonement measure for those witnesses who have spoken falsely in a court (but for a good purpose). But, even then they have to perform the atonement to regain purity that they have lost by speaking falsehood.
Yet another penance is the following:
8.106. Or such (a witness) may offer according to the rule, clarified butter in the fire, reciting the Kushmanda texts, or the Rik, sacred
to Varuna, ’Untie, O Varuna, the uppermost fetter,’or the three verses
addressed to the Waters.