Was there something more mystical about Mount Doom that meant another volcano wouldn't work and no matter how much energy was put into it, it would not be destroyed? A comment mentioned that not even Ancalagon the Black, the greatest dragon, could not have done it; but how about a Balrog? What about another Maia or a Vala? Could a sufficiently advanced technology destroy it? Given that middle earth was becoming more industrialized wasn't it only a matter of time until a furnace or other artificial heat source that was hot enough was created? If some catastrophe destroyed Middle Earth entirely would the ring be unmade, or would Eru have to do it himself?
A possible explanation from the canon is that volcanoes were remainders of the doings of Melkor (Morgoth) in his attempts to undo the work of the Valar in the "days before the reckoning of time". In the Valaquenta, the Silmarillion says of Melkor:
From splendour he fell through arrogance to contempt for all things save himself, a spirit wasteful and pitiless. Understanding he turned to subtlety in perverting to his own will all that he would use, until he became a liar without shame. He began with the desire of Light, but when he could not possess it for himself alone, he descended through fire and wrath into a great burning, down into Darkness. And darkness he used most in his evil works upon Arda, and filled it with fear for all living things.
And in "Of the beginning of days":
And in the darkness Melkor dwelt, and still often walked abroad, in many shapes of power and fear, and he wielded cold and fire, from the tops of the mountains to the deep furnaces that are beneath them; and whatsoever was cruel or violent or deadly in those days is laid to his charge.
Volcanoes are explicitly associated with Melkor in the Silmarillion as the remaining scars on the world after the Valar "repaired" the damage:
In that time the Valar brought order to the seas and the lands and the mountains, and Yavanna planted at last the seeds that she had long devised. And since, when the fires were subdued or buried beneath the primeval hills, there was need of light, Aulë at the prayer of Yavanna wrought two mighty lamps for the lighting of the Middle-earth which he had built amid the encircling seas.
I think that Sauron, being Melkor's most powerful disciple, would have forged the Ring in the fires of Utumno, if he would have had the opportunity:
The lands of the far north were all made desolate in those days; for there Utumno was delved exceeding deep, and its pits were filled with fires and with great hosts of the servants of Melkor.
However, Melkor's fortress was utterly destroyed after his downfall. Perhaps Mount Doom still held most residue or echoes of Melkor's power or malevolence.
Mount Doom was where the Ring was forged, so the implication was that only its fires could destroy it.
If any other odd volcano could do it, Gandalf wouldn't need to risk the Ring falling into Sauron's hands by sending Frodo into the middle of Mordor
I think that physics phenomena such as heat are irrelevant here. The Ring must be destroyed at the end of the Journey, by a combination of Magic and Fate. It is not only the intense heat of Mount Doom that destroys it, but the fact Mount Doom is an important feature of Middle Earth, back from an ageless time when... etc, etc.
A technological terror like the Death Star could not make a dint in the One Ring, merely because said weapon is neither ageless nor magical in nature. [Note: I mention the Death Star because it is what the original question was about, before it got edited].
Now, if one were to use the Force... :P
Supporting statement from Elrond (from the movie, unfortunately):
The Ring cannot be destroyed, Gimli, son of Glóin, by any craft that we here possess. The Ring was made in the fires of Mount Doom. Only there can it be unmade. It must be taken deep into Mordor and cast back into the fiery chasm from whence it came.
He never says "... Or any other fire of equivalent intensity".
The Ring could be destroyed only by the Fires of Mount Doom, because that is the place it was made. The lesser Rings of power could be destroyed by dragon-fire, as Gandalf mentions:
"Your small fire, of course, would not melt even ordinary gold. This Ring has already passed through it unscathed, and even unheated. But there is no smith's forge in this Shire that could change it at all. Not even the anvils and furnaces of the Dwarves could do that. It has been said that dragon-fire could melt and consume the Rings of Power, but there is not now any dragon left on earth in which the old fire is hot enough; nor was there ever any dragon, not even Ancalagon the Black, who could have harmed the One Ring, the Ruling Ring, for that was made by Sauron himself.
There is only one way: to find the Cracks of Doom in the depths of Orodruin, the Fire-mountain, and cast the Ring there, if you really wish to destroy it, to put it beyond the grasp of the Enemy forever."
I believe Aule the Smith could destroy it, since he taught Sauron everything he knows, in his forge in Valinor if you could convince the Valar to allow the ring to go there, and no one stole it, but that is outside the world of Arda, so Gandalf is correct that there is nothing in Arda that could destroy it except Mt. Doom.