Is it possible to re-enter the birth and death cycle of life after one has achieved Moksha/Nirvana ?
No. It's not possible.
After Moksha, a soul has become God by merging into the singularity of God. What God does, that soul does or what that soul does, the God does. Both are same.
The question rather should be,
Is it required to have a life after Moksha?
A soul with no desire left and who has conquered its Rajo-Guna (the mode of passion) will not have anything to do with the material world. Even that soul doesn't have to guide anyone, because tendency to guide or help is also a (selfless) desire.
All the entity come to manifested from unmanifested when the day of Brahma starts. Unmanifested is the actual nature of God, which is Nirguna (mode-less), Nirvikara (distorion-less) and Nirpeksha (non-relative). However with its own Yoga-maya, the unmanifested God can also take a manifested form of soul, superior than the others.
Presumably, your question is about the regular souls like us. When such soul realizes the God, it finally merges within the God's unmanifested form. It's like a water droplet merging into the ocean.
Can we say that when a new body is formed later, the same droplet is again attached? We can safely assume that it's gone for good!
But those who worship Me, giving up all their activities unto Me and being devoted to Me without deviation, engaged in devotional service and always meditating upon Me, having ﬁxed their minds upon Me, O son of Pṛthā – for them I am the swift deliverer from the ocean of birth and death.
One who understands this philosophy concerning material nature, the living entity and the interaction of the modes of nature is sure to attain liberation. He will not take birth here again, regardless of his present position.
The souls which are going to attain Moksha would enjoy within the present life as well:
When the embodied being is able to transcend these three modes associated with the material body, he can become free from birth, death, old age and their distresses and can enjoy nectar even in this life.
yes, soul can always enter cycle of life-death even after achieving Moksha. Have you studied Advaita Vedanta and Shudhadvaita(non-dual reality)? Human souls are nothing but fragments of the divine God, and if maya or ignorance or the instinct of I is removed from soul, soul is equivalent to God. Than what the hell caused the maya in very first place to delude divine God himself and appear in the form of infinite souls. It is avidya, that is forgetting true self. Moksha simply means established in true self(the non-dual reality). And life simply means forgetting the true self and having desire of anything other than true self. Kindly study teachings of buddha.
2) Life is permeated through suffering caused by desire, 3) Eliminate desire and you eliminate suffering and attain moksha.
Also, study Samkhya laid out by Kapila muni, which revolves around Purush and Prakriti,
The idea of evolution in Samkhya revolves around the interaction of prakṛti and Purusha. prakṛti remains unmanifested as long as the three gunas are in equilibrium. This equilibrium of the gunas is disturbed when prakṛti comes into proximity with consciousness or Purusha. The disequilibrium of the gunas triggers an evolution that leads to the manifestation of the world from an unmanifested prakṛti. Purusha is pure consciousness absolute, eternal and subject to no change. It is neither a product of evolution, nor the cause of any evolute. It is what is called the soul of every one of us.
Samkhya considers ignorance (avidyā) is regarded as the root cause of this suffering and bondage (Samsara). Samkhya offers a way out of this suffering by means of discriminative knowledge (viveka). Such knowledge, that leads to mokṣa (liberation), involves the discrimination between prakṛti (avyakta-vyakta) and puruṣa (jña)
But if later purush again forgets to discriminate between prakriti and purush(himself) when prakṛti comes into proximity with him(purusha), he will again manifest or fall in cycle of life-birth. So, only method of liberation is to always remain in introversion, that is nirvikalp samadhi or see everything with indifference
No Soul after achieving Moksha will enter the cycle of birth and death because he is free from each and every imperfection.
But still they can come on earth to guide souls towards the kingdom of God. Just for the purpose of helping more souls realize the absolute truth, they come to earth. Though in this life they know for what purpose they are on earth, they have already risen above Maya and various imperfections. They are free.