Are there any sins that are not forgiven by God?
The basic teaching of Hinduism and the scriptures of it is to not to feel guilt of past mistakes, and even the sinful of sinners can cross over the Ocean of Samasra and come out of past sins through spiritual knowledge.
Bhagavad Gita 4.36 Even if you are considered to be the most sinful of all sinners, when you are situated in the boat of transcendental knowledge, you will be able to cross over the ocean of miseries.
When one is situated on the boat of spiritual Gyana, Isvara Gyana, the spiritual knowledge burns all your Karmas into Ashes.
Bhagavad Gita 4.37: Just as heat of fire reduces wood to Ashes, the fire of self knowledge burns to ashes all Karma
This was the statement given by Krishna paramatma in Bhagavad Gita. From this we can understand, only way to come out of all Karmas is Spiritual knowledge !
Actually, there is one sin that is never forgiven directly by God. An offence against a Bhagavata is unforgiveable except by the Bhagavata himself.
The episode of Maharaja Ambarīṣa and Sage Durvāsā is best example in this regard. When Durvasa offended Ambarisha (a great devotee of Krishna who was accorded protection by Sudarshana chakra), by cursing him for breaking a fast without Durvasa's permission. His curse and actions cause Sudarshana chakra to chase Durvasa to all corners and no deva or even Lord Vishnu would protect him. Then based on advice from Vishnu, Duravasa sought forgiveness at the feet of Ambarisha. Only then Sudarshana chakra stopped chasing the sage.
You can read the entire story in Srimad Bhagavatam.
So, Bhagavata apachaara is a great sin that the Lord will never forgive.
As per PADMA PURANA: PATALAKHANDA: Chapter 8
Sri Rama said :
24-26. Sin is said to be of two kinds, viz. deliberate and nondeliberate. Deliberate is that which is done with an intention. The non-deliberate is that committed unintentionally. The deliberate sin is exhausted only after experiencing (the fruit of that sinful) deed. The other one would perish through remorse (प्रायश्चित). This is declared by the scriptural injunctions.
Many atonements (प्रायश्चित) are declared in scriptures for variois nondeliberate sins. Many prayaschitta are discussed here.
According to Pushpadanta, chanting Shiva Mahima Stotra will free you from all 'sins'. Take it from here.
एककालं द्विकालं वा त्रिकालं यः पठेन्नरः
सर्वपाप-विनिर्मुक्तः शिव लोके महीयते|
ekakālaṁ dvikālaṁ vā trikālaṁ yaḥ paṭhennaraḥ |
sarvapāpa-vinirmuktaḥ śiva loke mahīyate || 42 ||
Whoever reads this once, twice or thrice (in a day) revels in the domain of Shiva, bereft of all sins. (42)
The quantifier 'all' seems to indicate there is no sin which is unforgivable.
This question is misconceived. God does not interfere in the running of the universe. God does not ordinarily do anything. It is nature that does every thing.
Resorting to Prakrti, Nature, which is My own Power, I send forth again and again this multitude of beings that are without any freedom, owing to Nature's sway over them.
These activities do not in any way bind Me, because I remain detached like one unconcerned in their midst.
Under My direction and control, Nature brings out this mighty universe of living and non-living beings. Thus does the wheel of this world revolve.
God will not interfere in your life if you do not want God to interfere in your life by praying or worshiping God. The only thing God will do is to distribute the fruits of your karma.
Hence there is no need for God’s forgiveness for any sin.
What then is the Hindu idea of paapa or sin and how can one get rid of the effects of the sin?
The idea of paapa, translated as sin, is different from the Christian idea of sin. Christian theology says that sin is an offense against God. Hence there is need for forgiveness from God in Christian theology. Paapa is an offense against oneself. One damages one self through bad karma. So there is no point in asking for forgiveness from God.
So how can one reduce paapa or sin?
How to get rid of the effect of sins
Yudhisthira says,’…a perpetrated sin is expiated by auspicious acts, by publishing it wildly, by repentance, by alms-giving, by penances, by trips to tirthas after renunciation of everything, by constant meditation on the scriptures. Of all these, he that has practiced renunciation is believed to be incapable of committing sins anew. ‘
Mahabharata Santi Parva Section VII
There are two other points to discuss.
Instead of asking if there are any sins that God does not forgive, a more relevant question is if Hindus are condemned to eternal hell since that would be the implication of a sin that can never be gotten rid of.
The answer is that most Hindu Acharyas or teachers reject the idea of eternal hell.
Does God then have no role in reducing the effect of sin?
Nanda: "But how can we obtain God's grace? Has He really the power to bestow grace?"
Sri Ramakrishna (smiling): "I see. You think as the intellectuals do; one reaps the results of one's actions. Give up these ideas. The effect of Karma wears away if one takes refuge in God. I prayed to the Divine Mother with flowers in my hand: 'Here, Mother, take Thy sin; here take Thy virtue. I don't want either of these; give me only real bhakti. Here, Mother, take Thy good; here take Thy bad. I don't want any of Thy good or bad; give me only real bhakti. Here, Mother, take Thy dharma; here, take Thy adharma. I don't want any of Thy dharma and adharma; give me only real bhakti. Here, Mother, take Thy knowledge; here take Thy ignorance. I don't want any of Thy knowledge or ignorance; give me only real bhakti. Here, Mother, take Thy purity; here take Thy impurity. Give me only real bhakti [devotion].'"
The Gospel of Sri Ramakrishna, Visit to Nanda Bose's house,July 28, 1885