I want to know whether a yogi/self realized person is free from his actions?
Is this mentioned in the Gita?
Does this mean that he can do whatever he wants and not be accountable?
A Jivanmukta is not free from his fruits of his action. He does not care about the fruits of action.
Wise men, established thus in the unperturbed evenness of mind, abandon the fruits of action, free themselves from entanglement in the cycle of births and deaths, and attain to the state of freedom from all sorrow (liberation).
What does abandoning the fruits of action mean?
It means that he is indifferent to what happens to him.
'He is the true Jivanmukta, whose facial expression neither flushes nor fades under pleasure or pain and who subsists on whatever comes of its own accord.
The flushing is of course indicative of joy. The joy, which ordinary men derive from sandal-paste and other marks of hospitality, does not rejoice him (make him happy). "Fading' means depression. He is not depressed by any loss of wealth, .... or similar calamities, that may befall him. 'Who subsists on whatever comes of its own accord' means, who remains content with whatever goes to him in the shape of alms, etc.
Jivanmuktiviveka of Vidyaranya
A jivanmukti is free from karmic consequence, good or bad, and from suffering. That’s because karma is generated in the consciousness (some would call this awareness) not in the pure chit of the Self. If you wanted, you could say he/she does not engage in karmic activity, but ultimately the two are equivalent - he has stopped generating karma and is free.
In my understanding, a yogi is not necessarily enlightened (?)
I am using the terms where sat-chit-ananda would be translated reality-awareness-peace. I do not refer to chit when I say consciousness. Some reverse the terms consciousness and awareness.
When Brahman is realized, the non-clinging and destruction of the subsequent and previous karma respectively.
Mundaka Upanishad II.ii.8
Sundered are the knots of the heart, torn off are all his doubts, and the seeds of his karma disappear when the knowing of Brahman is obtained.