Let us first try to understand what moksha means. Moksha is commonly translated as complete liberation.
So what are we being liberated from? For us to be liberated, we must be bound right? So what are we bound by (Bhagavad Gita 14:5)
Sattva, rajas and tamas—these qualities, O mighty-armed Arjuna, born of Nature, bind fast in the body, the embodied, the indestructible!
So, basically Sri Krishna says that these gunas of Sattva (Goodness),. Rajas (Excitement) and Tamas (Laziness) bind and enslave the indestructible soul, which is embodied in the body.
So liberation is basically being liberated from these gunas. The one who is unattached, the one who is free of desire, free of the six enemies (kama, krobha, lobha, moha, madha, matsarya), He is liberated.
Such a man, who is thus liberated, exercises FREE WILL. so what is free will? People usuallly think that free will means being able to do what one wants, but many saints say that such will is not really free, but rather bound will. For example, when we decide to do something we want, if we look deep what is our reason for doing the same, it is usually either our desire to get something (influence of senses), fear of losing (influence of mind), hatred against some/something (influence of ego) or just plain 'if I dont do it now, I will not be able to do it later' (influence of attachment and greed). Then how can such action be called FREE ACTION.
So long as we are under control of mind and body attachment or ego-attachment, so long as we identify ourselves as 'X' name and 'Z' form, then we cannot claim to have free will, it will always be bound by nature and therefore it is actually BOUND will and not really FREE.
Thus the man who has been liberated, still performs action, but He is free of the bondages which bind him, in other words, He performs duty without attachment to fruits, as prescribed by Sri Krishna in Gita 2:51:
The wise give up the results of their actions and thus liberate themselves from the bondage of material birth and death.
Simply put, the realized being now performs the truly FREE action, FREE will, Free from all desires.
Now, does this mean that the soul attains a so called 'abode' of Divine, which is a metaphorical plane of existence? That is interpreted differently by different schools of thought, Some believe that there is no such plane that liberated souls go to, because liberation itself is free-ing oneself from birth and death, therefore one who attains liberation is freed from all body-attachment and is therefore becomes ONE with God or His TRUE Self or Sri Krishna or Lord Shiva (whatever you want to call that Ultimate GOAL of life). According to other schools, one is REBORN in a so-called LOKA where He indulges in eternal play or Leela with Bhagavan or God.
If you specifically mean to ask what does the 'soul' do after moksha, rather than what kind of actions does one do once he attains moksha. Then the soul really 'does' what has always been doing = Being the Witness. Actions only happen in the relative plane. The soul is always unacttached and unaffected. The soul remains existence-knowledge-bliss (Sat chit ananda) as it always was.
Shankaracharya's Shivoham Mantra:
Mano Buddhi Ahankara Chitta Ninaham
Nacha Shrotra Jihve Na Cha Ghrana Netre
Nacha Vyoma Bhoomir Na Tejo Na Vayu
Chidananda Rupa Shivoham Shivoham
I am not mind, nor intellect, nor ego,
nor the reflections of mind-stuff (chitta).
I am not the five senses.
I am beyond that.
I am not the ether, nor the earth,
nor the fire, nor the wind (the five elements).
I am indeed,
That eternal knowing and bliss, Shiva,
love and pure consciousness.
Basically one realizes that He is not the body, not the mind, not the ego, but the SOUL (or Atma or Brahman or Vishnu or Shiva or a part of Brahman or a member of God's kingdom based on your school of thought). The SOUL is the witness and is never attached to actions, nor the enjoyer of its fruits.
If you meant what happens to the one (individual consciousness) once it reaches abode of Vishnu or Shiva (ultimate consciousness), then this is what Sri Ramakrishna Paramahansa says about realization/moksha and what happens after:
Once a salt doll went to measure the depth of the ocean. It wanted to
tell others how deep the water was. But this it could never do, for no
sooner did it get into the water than it dissolved. Now, who was there
to report the ocean's depth? What Brahman is- cannot be described. In
samadhi one attains the knowledge of Brahman -- one realises Brahman.
In that state reasoning stops altogether, and man becomes mute. He has
no power to describe the nature of Brahman.
Basically from advaitic standpoint, nobody really knows what happens to individual consciousness once it merges in Absolute (once self-realization occurs or once one realizes that he is not the body, mind, ego, etc. but Atman/Brahman/Shiva/Vishnu/God). One can only speculate