Lying / false promise is a sin.
The God, Mother, Father and the Guru are considered equivalent.
As per some verses, Mother is the greatest. So fake oaths in reference to any of the above is sinful.
In any case, the results of false promise, and breaking vows are considered sinful and results in a torture in hell after death, and low-specie birth on Earth.
In reference to making fake religious oaths or promises, the results are quite grave. For instance -
Chapter 26 - The Importance of Keeping Promise, Brahma-khaṇḍa, Padma Purāṇa
O brāhmaṇa, the merit in (i.e. due to) keeping a promise is a crore
times greater than the fruit which a man obtains by giving a hundred
cows. There is no doubt that a fool by breaking his promise goes to a
terrible hell and is roasted there for a hundred periods of Manu. Then
due to his own deeds he obtains birth (i.e. is born) on the earth in
the house of a poor person, and bereft of food and garments he is
afflicted. One should take an oath truthfully in the vicinity of a
deity, fire, or one’s preceptor. Till that time the body is burnt; but
the family of Viṣṇu is not lost (i.e. continues). O brāhmaṇa, what
should I now say about a false oath? O brāhmaṇa, a man (lives) in hell
for a hundred periods of Manu due to a false promise.
O best sage, he
who touches the flowers taken off after Viṣṇu’s worship and takes a
truthful oath (but does not keep it), is roasted, taking with him
seven members (of his family), in hell for a long time. If by chance
he gets a birth, then in every existence (he becomes) a leper. This
would never happen due to truthful promise. What can be said about
(what would happen) due to a false oath? The man who offer his right
hand (to give a promise) and keeps it, would obtain Kṛṣṇa.
telling the truth, (and) the truth (only). The dead ancestors of him
who, having offered his hand (to make a promise), does not keep it,
are tormented in hell; and undoubtedly the dead one (i.e. he, after
death) goes to a terrible hell, and is emancipated by crores of men
(of his family, i.e. after a very long time).
There are certain circumstances when falsehood might be admissible, but some lies are always inadmissible.
Manusmṛti Verse 8.112
कामिनीषु विवाहेषु गवां भक्ष्ये तथेन्धने । ब्राह्मणाभ्युपपत्तौ च शपथे
नास्ति पातकम् ॥ ११२ ॥
There is no serious offence in swearing to women, or in connection
with marriages, fodder for cows, or fuel, or for the sake of a
Gautama Smriti (23.29-31).—
‘Some declare that an untruth spoken in marriage, during dalliance, in jest, or while one is in severe pain, is not reprehensible. But that is certainly not the case when the untruth concerns the guru; for if one lies even in his heart to his guru, regarding even small matters, he destroys himself, his seven descendants and seven ancestors.’
Read more at -
Circumstances sanctioning falsehoods
Finally, in the Mahabharata, Yuddhishthira comments as follows -
Section CIX, Rajadharmanusasana Parva, Shanti Parva
Sometimes men (robbers), desirous of obtaining the wealth of some
one, make enquiries (for facilitating the act of plunder). One should
never answer such enquiries. That is a settled duty. If by maintaining
silence, one succeeds in escaping, one should remain silent. If, on
the other hand, one’s silence at a time when one must speak rouses
suspicion, it would be better on such an occasion to say what is
untrue than what is true. This is a settled conclusion. If one can
escape from sinful men by even a (false) oath, one may take it without
incurring sin. One should not, even if one be able, giveaway his
wealth to sinful men.
To conclude -
Lying or false oaths/promises in reference to one's Guru, Parents and the God are unacceptable. They are sinful and results in hell and other karmic punishmenta.
Under some circumstances, especially in reference to protecting the Dharma, or in emergencies (āpaddharma), lying maybe permissible, and sometimes even meritorious.