The issue of what Gandalf knew concerning the destruction of the One Ring is discussed in the answers to How does Gandalf (or anyone) know how to destroy the Ring? As for the lesser rings, Gandalf states not only that these could be destroyed by dragons, but that four were in fact destroyed in this way.
Seven the Dwarf kings possessed, but three he [Sauron] has recovered,
and the others the dragons have consumed.
(The Shadow the the Past)
This is supported by the following quote from "Of the Rings of Power and the Third Age" (the Silmarillion):
It is said that the foundation of each of the Seven Hoards of the
Dwarf-kings of old was a golden ring; but all these hoards long ago
were plundered and the Dragons devoured them, and of the Seven Rings,
some were consumed in fire and some Sauron recovered.
None of this brings us any closer to answering the question: how did Gandalf know this? Unfortunately, we know next to nothing about the seven, except the ring of Durin III which was seized by Sauron when he captured Thrain II around TA2845. We do know that dragons reappeared on the Withered Heath around TA2570, and that the ring of Durin III was the last to be lost (see the entries in the Tale of Years). That gives us roughly a 300 year window in which four of the seven were destroyed by dragons. Gandalf arrived in Middle-Earth around TA1000, so he was present the whole time, and had plenty of time to investigate (the Council of Elrond took place in TA3018). It could be that he simply found eye witnesses. If the dwarf kings kept the rings on their person, then there are probably only two outcomes when a (fire-breathing) dragon invades:
Of course, if the ring was hidden away deep in the vaults when the king was incinerated then it's hard to see how Gandalf could have been so sure of its fate. There is another, less prosaic possibility. In Of the Rings of Power and the Third Age (the Silmarillon), we find the following:
But the elves were not so lightly to be caught. As soon as Sauron set
the One Ring on his finger they were aware of him...
(see also the Council of Elrond and the Tale of Years entry for (c) SA1600). Thus the bearers of the Three could sense the power of the One, and may similarly have felt some connection to the Seven, at least enough to sense their destruction.