To the best of my knowledge burning as a punishment is not associated with a furnace anywhere in Tanach. I am not aware of any parallels in ancient near eastern religions either. I would like to know if any commentaries explain why Nevuchadnezzar chose to consign those who chose not to bow down to his statue ie Chanaya, Mishael and Azarya to death by burning in a furnace specifically (Daniel 3:6, passim and see Yirmiyahu 29:22)
There are a few parallels or partial parallels in the Near East and in Tanach & Midrash:
Near East Judicial Processes:
"Execution by burning occurs in Mesopotamia both as a provision of the legal system for certain crimes and as a punishment imposed by the king. It is attested already in the Old Babylonian period. The Code of Hammurabi prescribes death by burning for three offenses: for thefts committed during a fire (the thief is cast into that same fire [CH 25])...The Old Babylonian omen CT 6, 2 (case 14) envisages the following situation..."the high priestess will steal the taboo offering, but she will be captured and burned...a letter from the archives of Mari, contemporary with the preceding text, rhetorically alludes to death by burning simply for harboring forbidden thoughts...Burning occurs also in later sources from the first millennium...[in a Babylonian judicial chronicle from the Hellenistic period]...three cases of sacrilege are reported...The punishment involved is almost invariably death by fire..."
Beaulieu points out that from these texts one cannot tell in which manner these criminals were burned. Later in the essay, he writes:
"...It is fortunate that we have three instances in Mesopotamia where the manner of execution by burning is specified, and all three cases involve being thrown into an oven or a furnace...The earliest text (BIN 7, 10) is a letter of King Rim-Sin of Larsa...:
"Speak to Lu-Ninurta, Balmu-namhe, Ipqu-Erra and Mannum-kima-Sin: Thus says Rim-Sin, your lord. Because he cast a boy into the oven, you, throw the slave into the kiln"
...The two words for "oven" and "kiln" are tinuru and utunu. The latter derives from the Sumerian Udun, and occurs more rarely as atunu, the form under which it entered the Aramaic language (אתון in Daniel 3). The second occurrence comes from a palace edict of the Assyrian king Assur-resa-isi...:
"...she is veiled for her lord. They shall throw them, either the woman or the man, the eye-witness, in the oven."
...The third and final example occurs in a Neo-Babylonian school text from the Sippar temple library...:
"To Enlil-naidn-sumi, governor of the land...son of the loins of princeship, superintendent of all [the cult centers of A]kkad, speak, thus [Samsu-ilun]a, king of the world...Inwardly they profane and desecrate their god...You now, destroy them, burn them, roast the...to the cook's oven...make their smoke billow, bring about their fiery end with the fierce flame of the box-thorn1!"
Ashurbanipal, king of Assyria and identified by some with Ossnapar from Ezra, had a tablet made in which he described his battles against his brother Shamash-Shum-ukin, also known as Saulmugina. When Shamash-Shum-ukin was finally captured, he was tossed into a fire:
"...Saulmugina my rebellious brother, who made war with me; in the fierce burning fire they threw him, and destroyed his life..." (source, pg. 374)
Ashurbanipal also had another rebel named Dunanu burned by furnace (source).
Note: Several Christian sites online state that some sort of kiln was discovered in digs in Babylon and was inscribed with the following: "this is the place of burning where men who blasphemed the gods of Chaldea died by fire." However, I have yet to find a more reputable source for this.
Near East Fire Worship:
- In Mesopotamia there was a ritual called "Šurpu". The purpose of the ritual was a kind of purification from sin, which involved confession and the subsequent burning of an object such as red wool, garlic or onion peels. It was thought that during the ritual, the confessional process transferred the sins of the repenter to the objects that was burned.
There were a few Babylonian fire deities:
- Girra, god of fire, light and metal. A ritual to counter witchcraft which includes invoking Girra also makes mention of a furnace (albeit, as a metaphor):
"...may their witchcraft...turn upon them...May Girra scorch them, may Girra melt them, may Girra fire them...Smash them like a pot and like a furnace may their smoke cover the heavens..." (Source)
Later on in history, the three were mixed and matched. It's possible that the statue Chananyah, Mishael and Azaryah refused to bow down to was of one of these three deities, making their punishment all the more fitting.
Nimrod, most famously, threw Avraham into a pyre or a furnace (depending on which version of the midrash - Beresheet Rabbah 38:13, fragments of Midrash Avraham and Nimrod (can be seen on the site of the Genizah or here), Eruvin 53a, Beresheet Rabbah 44:7, etc). There are some of the more secular scholars who see this act, together with Nimrod's title "gibor", as a sign of Nimrod being the forebear of Zoroastrianism, an ancient Persian religion in which fire is a central motif and whose worshippers were called "gabrs" or "gabers".2
Molech was worshiped in a manner similar to the usage of a furnace. The midrash in Tanchuma Va'etchanan 2:2 says:
"And how [did he do it]? Molech was in the valley of Ben-hinnom. It took place outside of Jerusalem and in a remote place. There was also an image [there] with the face of a calf and with its hands extended like a human [whose hands are open to receive] something from one's friend. Then they heated it until its hands became like fire. It also had seven latticed gates, with [the image] behind the innermost of them. Each and every one would enter in accordance with his offering. Whoever offered a bird entered the first lattice gate. With a goat he entered the second lattice gate; with a lamb, the third; with a calf, the fourth; with a bullock, the fifth; with a bull, the sixth. To whoever was offering his child the idol priests would say that there is none higher than that. He entered within the seventh lattice gate. Then he went and kissed it, as stated (in Hos. 13:2): THOSE WHO SACRIFICE A HUMAN BEING KISS CALVES. Then the idol priests would take his child from him and put the young child upon the hands of Molech, while they took the drums and beat on them, so that the father would not hear the voice of his child. Then they pressed the young child's entrails upon it, while the young child screamed until [its spirit] left its body in the hands of [the idol]."
Based on this description, the Molech was thought to look something like this: (picture from Wikipedia)
The death of Achav ben Koliah and Tzidkiyyah ben Ma'asiyah:
"Thus said the LORD of Hosts, the God of Israel, concerning Ahab son of Kolaiah and Zedekiah son of Maaseiah, who prophesy falsely to you in My name: I am going to deliver them into the hands of King Nebuchadrezzar of Babylon, and he shall put them to death before your eyes. And the whole community of Judah in Babylonia shall use a curse derived from their fate: “May God make you like Zedekiah and Ahab, whom the king of Babylon consigned to the flames!”— because they did vile things in Israel, committing adultery with the wives of their fellows and speaking in My name false words which I had not commanded them. I am He who knows and bears witness—declares the LORD."
On this the gemara in Sanhedrin 93a says:
"It is not stated: Whom the king of Babylonia burned, but “whom the king of Babylonia toasted.” Rabbi Yoḥanan said in the name of Rabbi Shimon ben Yoḥai: This teaches that he rendered them like toasted wheat, which is toasted on all sides."
On this the Ben Yehoyadah writes:
"מְלַמֵּד שֶׁעֲשָׂאָם כִּקְלָיוֹת. נראה לי בס"ד הכונה אם היה זורקם לתוך האש לא היו נעשים כקליות אלא היו נעשים אפר וגחלים ותתאבד צורתן אך הוא הניחם בתוך מחבת של נחשת והעבירם בתוך הכבשן כאדם שתולה בשר בתוך המחבת ומאת ה' היתה שהניח בלבו לעשות כן כדי שיוציאום אחר כך מן הכבשן ויהיו נראין קלויים בעוד צורתן עליהם לעיני הכל ויתבזו ויתפרסם הנס ביותר, מה שאין כן אם היה זורקן לכבשן היו נעשים אפר ואין ניכרים ולא היה אפשר להוציאם קודם שיהיו אפר וגחלים כי מי יוכל להתקרב לכבשן להוציאם אבל במחבת יש ידות ארוכות ושלשלאות ברזל קבועים בה ומשכום והוציאום אחר שנקלו."
Translation: This teaches that he rendered them like toasted wheat. It seems to me BS"D that the meaning is that if he would have thrown them into the fire they wouldn't have become like toasted wheat, but would have become ashes and coals and their shape would have been lost, but he placed them inside a copper pan and passed them through the furnace like a man who hangs his meat inside a pan and from Hashem was this that he placed inside his heart to do so, so he'll take them out of the furnace afterwards and they would look toasted while their shape shall hold before the eyes of all and they shall be embarrassed and the miracle shall be made famous, which would not have been had he simply thrown them into the furnace they would have become ashes and not noticeable and it would not have been possible to remove them prior to them becoming ashes and coals for who can come close to a furnace, but a pan has handles and iron chains attached to it and they pulled them out once they were toasted.
In conclusion, I think that considering all of the evidence, burning the three by furnace-fire was not such a random, rare event (unfortunately, of course, for the people who did die in that manner).
1 Interestingly, boxthorn is the אטד plant which is mentioned a few times in Tanach.
The Malbim notes:
זה עונש לממרה מצות המלך, וגם לשרוף ולבער את המורד מן העולם שיהיה כלא היה ולא תעשה מרידתו רושם בהענין הרוחני אשר כיון במלכות הנצחי
This punishment is for the embitterment of the King's command (i.e. for failing to bow down to this graven image), and also to burn and destroy anyone who rebels from the world and make it as if it had never been, and to not do any form of rebellion that makes an impression in spiritual matters since (he viewed it as threatening what he thought of as) the eternal kingdom.
Therefore, this Malbim would seem to suggest that he wanted to employ burning as a means to completely eradicate and leave no trace of any would-be rebel that threatened his kingship and that is why he specifically chose the punishment of burning.
Indeed such was his anger that the Midrash Shocher Tov 28:2 relates that he raged, "If the furnace is usually heated with one bundle, let it now be filled with seven, and if usually lit with seven, let it now be lit with forty-nine!" - So it is clear that he was incredibly annoyed by these three men who stood in his way and he wanted to completely decimate them!
It is worth noting parenthetically the flip-side that it is clear from a number of sources that they reasoned that they had to specifically go through this furnace for a number of reasons.
Firstly, it is worth noting the Gemara in Pesachim 53b where they learnt out a kal vachomer from the Frogs in the second plague who jumped into the ovens:
נשאו קל וחומר בעצמן מצפרדעים ומה צפרדעים שאין מצווין על קדושת השם כתיב בהו ובאו [ועלו] בביתך [וגו׳] ובתנוריך ובמשארותיך אימתי משארות מצויות אצל תנור הוי אומר בשעה שהתנור חם אנו שמצווין על קדושת השם על אחת כמה וכמה
They drew an a fortiori inference on their own from the plague of frogs in Egypt. With regard to frogs, which are not commanded concerning the sanctification of the name of G-d, it is written: “And the river shall swarm with frogs, which shall go up and come into your house, and into your bedchamber, and onto your bed, and into the houses of your servants, and upon your people, and into their ovens and kneading bowls” (Exodus 7:28). When are kneading bowls found near the oven? You must say that it is when the oven is hot. If in fulfilling the command to harass the Egyptians, the frogs entered burning ovens, all the more so, we, who are commanded concerning the sanctification of the name of G-d, should deliver ourselves to be killed in the fiery furnace for that purpose. (Sefaria translation and notation).
Indeed the Gemara in Kesubos 33b notes that they had to endure burning rather than the punishment of lashes:
דלמא מלקות חמור דאמר רב אילמלי נגדוה לחנניה מישאל ועזריה פלחו לצלמא
Perhaps the punishment of lashes is more severe, as Rav said: Had they flogged Hananiah, Mishael, and Azariah (see Daniel, chapter 3) instead of casting them into the fiery furnace, these three would have been induced to worship the graven image. (Sefaria translation and notation)
Furthermore the Sifrei Devarim 306 notes that Chananya, Mishael and Azaryah descended into the furnace so that Hashem could make nissim (miracles) for them in order that His name be sanctified in this world.