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'A red-hot axe was placed before two persons accused of theft charge and both were asked to touch the axe one by one. The thief got his hand burnt while the other one remained unhurt'

the above is quoted in the Bengali version of the Vivekachudamoni by Swami Dhireshananda (Udbodhan, page 228).

The process is said to have its origin in the Chandogya Upanishad (6/16/1-2), while such processes seem to be Tantrik to me.

Can someone identify the source and explain the process in avalable detail ?

The Upanishad passage you mentioned is from Chandogya Upanishad's 6th Adhyaya 16th Khanda:

VI-xvi-1: ‘Dear boy, (The officers of the king) bring a man, holding
him by the hand (while saying) "He has taken something, he has
committed a theft, heat the axe for him". If he is doer of that, then
he makes himself false. And being addicted to falsehood, he covers
himself with falsehood and grasps the heated axe; he is burnt, and
then he is punished.

VI-xvi-2: ‘If, however, he is not the doer of that, then he makes
himself true. And being attached to truth, he covers himself with
truth and grasps the heated axe; he is not burnt and then he is
released.

VI-xvi-3: ‘And as in this case he (the man attached to truth) is not
burnt, (similarly a man of knowledge is not born again). Thus has all
this world That for its self. That is the true. That is the Atman.
That thou art, O Svetaketu.’ From his words Svetaketu understood That
– yea, he understood

There are the processes of "test by water" and "test by fire" to test whether someone is claiming falsely or not. I think this should work for the kind of situation you have mentioned.

See the following verses:

Manu Smriti 8.114. Or the (judge) may cause the (party) to carry fire
or to dive under water, or severally to touch the heads of his wives
and children.

8.115. He whom the blazing fire burns not, whom the water forces not to come (quickly) up, who meets with no speedy misfortune, must be
held innocent on (the strength of) his oath.

8.116. For formerly when Vatsa was accused by his younger brother, the fire, the spy of the world, burned not even a hair (of his) by reason
of his veracity.

So, I think if the accused person is made to touch fire and if he does not get burnt in the process, then it is to be concluded that he is innocent.

The Yajnvalkya Smriti's 2nd Chapter gives more such Divine tests to ascertain the innocence of the accused persons. There are the Ordeal of Tula (balance), that of Fire, that of water, poison etc.

Tula (weighing in the balance), Agni (fire), Jala (water), Visha
(poison) and Kosha, are the ordeals (laid down for establishing the
innocence of the accused) ... (97)

• Ordeal of Tula (balance):

The accused after sitting on the scale of a balance, should have
himself weighed by an expert in weighing,(such as a goldsmith, etc,)
(with clods of earth or stones) equalling in weight, then marking (the
measured weight), he should get down (from the scale). (102)

(He should then declare) "O Balance! thou art the abode of Truth,
formerly thou hadst been made by the Celestials O thou of
auspiciousness, do thou speak the Truth and free me from suspicion.
(103)

If I perpetrated the crime, O mother, do thou bring me down, (making
me heavier than the weight); if I am pure, do thou take me up (i.e.
make me lighter)" (The accused should) inspire the Balance with the
above Mantram. (104)

• Ordeal of Fire:

Having marked (with any dye), the wounded parts etc, of) the palms
that had ground rice, one should place there seven fig leaves and
encircle them with an equal number of threads. (105)

"O Fire, O purifier, thou dost range in the hearts of all creatures;
Thou art, O Kavi, the witness of virtue and sin, do thou speak out
Truth relating to me" (106)

(After the accused) had recited (this Mantram, the Judge) should place
on both the palms (of the accused, two) redhot even (iron) balls, of
the size of fifty palas (each). (107)

Having taken them, he (i.e. accused) should gradually pass through the
seven Mandalas (circles), each circle is to consist of sixteen fingers
in extent and is to be severally placed at an equal distance (i.e. of
sixteen fingers each) (108)

If after having thrown off the burning iron balls and ground rice, (it
is seen that, the palms) are not burnt, (the accused) establishes his
innocence. If the balls are thrown within the limit or any suspicion
arises, (the accused), must, again, have to undergo the ordeal. (109)

Similarly, there are a few other such tests as I have mentioned above.

• Ordeal of Water:

Having inspired the water with the Mantram "O Varuna, protect me with
Truth" and held a person, standing navel-deep in the water, (the
accused) should drown himself in it. Simultaneously an arrow should be
discharged (from that place) and a strong man should be despatched
(where it falls and be made to return with the shaft) (If after his
return, he sees the accused) underneath the water, it establishes his
innocence. (110-111)

• Ordeal of Poison:

"O Poison! thou art the son of Brahma, established in the practice of
Truth. Save me from accusation, displaying the Truth, be like ambrosia
unto me " (112)

Having recited (this Mantram, the accused) should drink the Poison
formed on the summit of the Himalaya. His innocence is established,
who survives (the ordeal) without undergoing the least physical
change. (123)