Suppose a person is doing a job (say farmer or acharya/professor etc.,).
If (s)he becomes jivan mukta by sadhana then is (s)he bound to do his karmic activity as usual or has to leave the profession or it is up to the will of the person?
I think it would be better to ask whether the person who attains to jivamukta has a continuation of their Prarabdha karma. If a person is a professor, is it their prarabdha karma to continue as a professor? If their prarabdha karma is to continue as professor, then they will continue as a professor. Remember, however, that a jivamukta is entirely identified with Brahman, so it all continues as a dream to them. Sankara addresses the effects of prarabdha karma in his Aparoksanubhuti (Swami Vimuktananda translator):
The theory one hears from the scripture that Prarabdha does not hold upon one even after the origination of the knowledge of Atman, is now being refuted.
From the scripture--From such scriptural texts as: "The delay in his case os only so long as he is not released (from the body), then he will attain Brahman" (Chhandogya Upanishad 6.14.2).
Prarabdha does not lose its hold, etc.--The Shruti in many places has declared that even a Jnani is not free from the operation of Prarabdha. Shankara has dealt with this point at length in his commentaries on Chhandogya Upanishad (6.14.2), Vedanta Sutras (4.1.15), and Gita (4.37). In all those places he has supported the popular view that Prarabdha is binding on even on the Jnani. But here as well as in his Vivekachudamani (453-463) he has boldly asserted the true Vedantic view without any compromise. He has clearly shown that to a Jnani there is no such thing as a body, and it is meaningless to say that he is any longer under the influence of Prarabdha, which has no hold upon the bodiless Atman. The author brings his arguments in support if this view in stanzas 91 and 92.
After the origination of the knowledge of Reality Prarabdha verily ceases to exist, insomuch as the body and the like becomes non-existent; just as a dream does not exist on waking.
[translator's commentary] Insomuch as the body etc.--The body, mind, intelligence and the like have their existence only in ignorance and therefore cannot exist when the latter is entirely destroyed by Knowledge. In the absence of the body, Prarabdha also necessarily ceases to exist, since there remains nothing on which it can act.
And Brahma Sutras 4.1.15 (Swami Vireswarananda translator, https://www.wisdomlib.org/hinduism/book/brahma-sutras):
Adhikarana summary: Works which have not begun to yield results are alone destroyed by Knowledge and not those which have already begun to yield results
अनारब्धकार्ये एव तु पूर्वे, तदवधेः ॥ १५ ॥
anārabdhakārye eva tu pūrve, tadavadheḥ || 15 ||
anārabdha-kārye—Works which have not begun to yield results; eva—only; tu—but; pūrve—former works; tadavadheḥ—that (death) being the limit.
- But (of his) former works only those which have not begun to yield results (are destroyed by Knowledge); (for) death is the limit (set by the scriptures for Liberation to take place).
In the last two topics it has been said that all the past works of a knower of Brahman are destroyed. Now past works are of two kinds: Sanchita (accumulated) i.e. those which have not yet begun to bear fruit, and Prarabdha (commenced) i.e. those which have begun to yield results, and have produced the body through which a person has attained Knowledge. The opponent holds that both these are destroyed, because the Mundaka text cited says that all his works are destroyed. Moreover, the idea of nonagency of the knower is the same with respect to Sanchita or Prarabdha work; therefore it is reasonable that both are destroyed when Knowledge dawns.
The Sutra refutes this view and says that only the Sanchita works are destroyed by Knowledge, but not the Prarabdha, which are destroyed only by being worked out. So long as the momentum of these works lasts, the knower of Brahman has to be in the body. When they are exhausted, the body falls off, and he attains perfection. His Knowledge cannot check these works, even as an archer has no control over the arrows already discharged, which come to rest only when their momentum is exhausted. The Sruti declares this in texts like, “And for him the delay is only so long as he is not liberated (from this body); and then he is one (with Brahman)” (Chh. 6. 14. 2). If it were not so, then there would be no teachers of Knowledge. Therefore the Prarabdha works are not destroyed by Knowledge.
What the scriptures say is that such person can not gain fresh fruits of Karmas even if he or she engage oneself. That means a truly self-realised person can do Karmas but he still will not be affected by them. Whatever Karmas he had, both bad and good, are permanently destroyed (upon self-realisation or attaining Jivanmukti) and he can not gain fresh ones even if he keeps doing them.
The first verse is from Yoga VAshishta SAra's 3rd Chapter:
Sphatikah prativimbena yathA nAyAti ranjanam |
Tajagyah karmaphalenAnta stathA nAyAti ranjanam ||
Just as the clear crystal does not get (actually) colored by reflecting the colors, a Tattva-GyAni (Self-realised), in a similar manner, does not get affected by the fruits of his actions (karmaphala).
Yoga-VAsishtha-SArah, TattvagyAna-prakarana, verse 8.
Yet another verse from the same scripture:
SamAdhimatha karmAni mA karotu karotu vA |
HridayenAsta-sarvAsho mukto evottamAshayah ||
Whether the high-souled, self-realised person does any Karma or not in either cases he is always free.
Verses having similar purport are found in Tantras too. Here are a couple of them:
KriyamAnAni karmAni jyAn-prApteranantaram |
Na cha sprishati tattvagyam jalam padmadalam yathA ||
Just as the water drops can not stay on the lotus leaf in a similar way the self-realised person does not get affected by whatever Karmas he does.
KulArnava Tantram 9.127
Tattva-nishthasya cha karmAni punyApunyAni samkshayam |
PrayAnti naiva lipyante kriyamAnAni vA punah ||
For the the Tattva-nishtha person, all Karmas (Punya nad PApa) are destroyed. Even if he does those acts again, he does not get affected by them.
So, the answer is "Yes he can continue".
Yes, he can continue his profession, because bondage and liberation is of mind alone, not the body nor Atman.
When Brahman takes an avatar i.e. Saguna Brahman, it is jivanmukta by default and they perform duties/Dharma to teach mankind as they perceive everything as Self/Brahman.
Bhagavad Gita 3.22 There is no duty for me to do in all the three worlds, O Parth, nor do I have anything to gain or attain. Yet, I am engaged in prescribed duties.
Bhagavad Gita 3.24 If I ceased to perform prescribed actions, all these worlds would perish. I would be responsible for the pandemonium that would prevail, and would thereby destroy the peace of the human race.
Bhagavad Gita 3.25 As ignorant people perform their duties with attachment to the results, O scion of Bharat, so should the wise act without attachment, for the sake of leading people on the right path.
Bhagavad Gita 3.26 The wise should not create discord in the intellects of ignorant people, who are attached to fruitive actions, by inducing them to stop work. Rather, by performing their duties in an enlightened manner, they should inspire the ignorant also to do their prescribed duties
Jivan Mukti and Videha Mukti are explained by Lord Rama to Hanumana
I-i-15-17. Hanuman: Rama, sages speak differently: some say there is only one kind of liberation. Others say it can be got by worshipping your name and by the Taraka mantra at Kashi. Others speak of Sankhya-Yoga and Bhakti-Yoga, the enquiry into Vedanta-Vakyas etc.
I-i-18-23. Rama: Liberation is of four kinds: Salokya, Sampeepya, Sarupya, Sayujya etc. But the only real type is Kaivalya. Anybody even though leading a wicked life, attains Salokya, not other worlds, by worshipping my name. Dying in the sacred Brahmanala in Kashi, he will get the Taraka-mantra and also liberation, without rebirth. On dying anywhere (else) in Kashi, Maheshvara will utter the Taraka-mantra in his right ear. He gets Sarupya with me as his sins are washed away.
The same is called Salokya and Sarupya. Persevering in good conduct, with mind fixed upon me, loving me as the Self of all, the twice-born gets nearer to me – This is called the three forms of liberation. Salokya, Sarupya and Samipya
I-i-26-29. But by what means is the Kaivalya kind of Moksha got ? The Mandukya is enough; if knowledge is not got from it, then study the Ten Upanishads. Getting knowledge very soon, you will reach my abode. If certainty is not got even then, study the 32 Upanishads and stop. If desiring Moksha without the body, read the 108 Upanishads.
I-i-40-43. These destroy three kinds of Bhavana (regarding) body, senses and mind respectively as identified with Atman. The best of Brahmanas will become Jivanmuktas if they study upto the destruction of Prarabdha, these 108 Upanishads from a Guru along with the Shanti-pada. Then, in course of time they will get Vedeha-mukti, surely.
II-i-1. Then Hanuman asked Ramachandra: What is this Jivanmukti, Videha-mukti ? What is the authority, means of success and purpose ? Rama said: For a person there is bondage from doer-ship, enjoyer-ship, pleasure, pain etc., -- their prevention is liberation in the body. Videha-mukti (liberation without the body) is from the destruction of Prarabdha (operative) karma, like the space in a pot released from the conditioning (enclosing) pot. For both the authority is the 108 Upanishads. The purpose is eternal happiness through the stopping of the misery of doer-ship etc. This can be achieved by human effort just as a son is got by putra-kama sacrifice, wealth by trade etc., and heaven by jyotistoma.
II-ii-16. The wise know that the mind is bound by the impressions, it is liberated when released well from them. So, O Hanuman, practice the destruction of mental impression, quickly.
II-ii-17-18. When impressions die out, the mind becomes put out like a lamp. Whoever gives up impressions and concentrates on Me without strain, he becomes Bliss.
II-ii-19-23. Whether he concentrates on actions or not, when he avoids all the desires of the heart, he is doubtlessly liberated. He has nothing to gain from action or inaction. If his mind is not freed from impressions even Samadhi and Japa cannot give fruit. The highest place cannot be got without silence free from impressions. Sense organs like the eye go towards external objects without voluntary impression but because of the latent impression just as the eye falls voluntarily without attachment on external objects, so the man of wisdom operates in work.
II-ii-32-37. In a Jivanmukta the destruction of the mind is with form – in a Videhamukta it is formless – when you achieve it, the mind with qualities like friendliness will surely attain peace. The mind of the Jivanmukta has no rebirth.
Basically, a jivanmukta stops forgetting past births even with the death of temporary body and is established in his real self/Soul as also mentioned by Lord Krishna to Arjuna, while Videha Mukta disappears in Brahman like a salt doll in ocean. Infact, the entire Sanatana Dharma is based on the concept of Atman(Soul),its realization, reincarnation and laws of Karma which differs it from modern Abrahmic religions where the presence of soul is rejected and body alone is considered only reality and worshipped. Their had been many Jivanmuktas like Buddha, Adi Shankaracharya, Ramakrishna Paramahansa, Ramana Maharshi, King Janaka, Ashtavakra, Yagyavalkya, Uddalaka and other sages etc., performing their duties in detached manner and as Karma Yogis for the welfare of society, but they are rare as one in million.
Bhagavad Gita 4.4 Arjun said: You were born much after Vivasvan. How am I to understand that in the beginning you instructed this science to him?
Bhagavad Gita 4.5 The Supreme Lord said: Both you and I have had many births, O Arjun. You have forgotten them, while I remember them all, O Parantapa.
Bhagavad Gita 7.3 Amongst thousands of persons, hardly one strives for perfection; and amongst those who have achieved perfection, hardly one knows me in truth.
One who attains REALISATION, will continue to discharge his duties as earlier, till the prArabdha for the body is completed.
That is where Karma Yoga comes into picture. A Realised person will be a TRUE karma yogi.
Following is the excerpt from Sri Ramana Maharshi's life.
Mr. Rangachari, a Telugu pandit in Voorhees' College of Vellore, asked about nishkama karma. There was no reply. After a time Sri Bhagavan [Ramana] went up the hill and a few followed him, including the pandit.
There was a thorny stick lying on the way which Sri Bhagavan picked up; he sat down and began leisurely to work at it [sic]. The thorns were cut off, the knots were made smooth, the whole stick was polished with a rough leaf. The whole operation took about six hours. Everyone was wondering at the fine appearance of the stick made of a spiky material.
A shepherd boy put in his appearance on the way as the group moved off. He had lost his stick and was at a loss. Sri Bhagavan immediately gave the new one in his hand to the boy and passed on.
The pandit said that this was the matter-of-fact answer to his question.
I am assuming jivanmukta means knowledge of Brahman
From Tales and Parables of Ramana Maharishi
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
As seen above, only in Samadhi one attains knowledge of Brahman. Like Ramana Maharishi you might be help out the shepherd as described in one of the answers, but that would mean you're in an elevated/higher state, let's say 90% or 99% where 100% is knowledge of Brahman.
So to your question answer would be at No because jivanmukti is only at samadhi.