Temperature means little by itself - portions of the Earth's
thermosphere reach 1500C, but the Ring wouldn't get that hot if it were
exposed to the thermosphere; the latter is far too thin. Lava would be
good at melting things partly because of the temperature, but also
because with its much greater density and higher specific heat the lava
is going to impart much, much more heat to a metal Ring in contact with
it than is a mixture of acetylene and air at sea level - even though
the lava's temperature is lower.
One can, I think, safely assume that dragons are nowhere near 3000C
inside, because if they were they'd melt or combust anything they
touched - that heat has got to go *somewhere*, and a lot of it will be
radiated no matter how well insulated the dragon is (so Smaug could
never sleep atop a heap of coins, he'd melt right through it). Also,
their blood, for instance Glaurung's, would be much hotter than the
evidence indicates if they were that hot inside. It's a safer bet that,
while they're hotter than ordinary animals inside, the fiery breath is
a result of a chemical reaction occurring at or near the point of
exhalation.
Also, we have no reason to think the Dwarves or anyone else has the
technology to melt iron. For much of human history we have lacked the
technology for doing so and used other techniques for ironworking -
primitive smelters and forges couldn't exceed a temperature of maybe
700C. Smelting is a process of reducing iron oxide and removing the
impurities, and the latter was done by hammering them out of a "bloom"
(a spongy and still very impure mass left after the iron in the ore
became semiliquid and much of the non-iron burned off) rather than
simply putting ore over a fire and watching the pure liquid metal run
out the way one could do with, say, copper ore. So Ancalagon's fire
could have been hotter than anything the Dwarves had and yet still not
hot enough to melt relatively pure iron, even if his breath could be
sustained and contained in a furnace.
It's probably also safe to assume that mithril requires more than just
extra heat to work it, as does, say, platinum, which wasn't
successfully isolated and worked until almost 1800 (and even then only
by using very dangerous, highly toxic chemicals). It may be that most
dragons' breath couldn't melt mithril alloys either (we know that the
Dwarves had dragon-resistant armor at the War of Wrath). The lava of
Orodruin may have been more than just hotter, it may have contained
and/or outgassed unusual chemicals which lowered the melting point of
the unique Ring alloy.
What this boils down to is that, even absent any supernatural
explanation, the lava at the Crack of Doom may have been "only" around
1000-1200C and still be hotter than any furnace in Middle-earth, yet
cooler than the breath of an above-average dragon, but even so still be
the only source of the fire and unique chemical properties that made
melting and/or working the Ring possible.
- Bruce Tucker
>> Stay informed about: Melting rings with dragon-fire 
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. iirc, Bilbo