What happened to the Seven Dwarven Rings? We have this:
- "He knows that it is not one of the Seven, or the Nine, for they are accounted for."
- "Seven the Dwarf-kings possessed, but three he has recovered, and the others the dragons have consumed."
Seven Rings of Power were created for the ensnarement of the Dwarves; four were destroyed and three were with Sauron at the end of the Third Age.
- "The Seven are lost to us - if Balin has not found the Ring of Thrór, which was the last; naught has been heard of it since Thrór perished in Moria. Indeed I may now reveal that it was partly in hope to find that ring that Balin went away."
- "Balin will find no ring in Moria, said Gandalf. Thrór gave it to Thráin his son, but not Thráin to Thorin. It was taken with torment from Thráin in the dungeons of Dol Guldur. I came too late."
Smaug doesn't get Thrór's Ring; Sauron aka the Necromancer reclaims it in the pits of Dol Guldur from Thráin, although this isn't known to the Dwarves until Gandalf informs them during the Council of Elrond. So the Dwarves had some knowledge of the disposition of the Seven, though that knowledge wasn’t complete. Although the secretive folk may not have announced to the rest of the world where the Seven may be stowed, seemingly some Dwarves knew.
- "It is 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 it was made by Sauron himself."
A dragon could, in theory, melt/consume a Ring of Power, excepting the One.
Regardless, we have four that aren't reclaimed, and if Sauron can't find something*, you can pretty sure that it's no longer available. This means that four were indeed melted or consumed by Dragons. I can't see the Dwarves feeding the Rings to the Dragons, but more likely the Dragons biting and eating the five-fingered hand that bore the Ring - that or completely cooking the Dwarf that held the Ring on one of its five fingers (assuming that Dwarves have five digits). The Dragon would then have to destroy the Ring completely, as a Dwarf wouldn’t leave the valuable item lying about - or sit on it like a hen to keep Durin’s folk from having it. If it ate the Ring, I would assume that the Dwarves would make a go of performing some type of medical procedure via an axe to see if the Ring remained with the gullet of the Worm (assuming that Dragons had gullets).
Also it can be assumed that these four Rings weren't born by your average run-of-the-mine Dwarf, but some Dwarf-lord with quadra-forked beard.
So, to recap: Four rings, four Dwarf-lords, four Dragons of some repute. Why then do we not read anything about the massive battles that obviously must have taken place in each of these four cases?
*Unless hidden by a witless hobbit.