There is little evidence that Faerun's Dwarvish and Elvish are significantly inspired by Tolkien.
Much about the dwarves and elves of Dungeon's and Dragons was heavily inspired by JRR Tolkien, though they are not entirely lifted from his pages. That they each have their own languages distinct from common is a feature of both Tolkien's elves and dwarves and the 1e D&D rules (and all subsequent editions of D&D, as far as I know). So Greenwood's giving the elves and dwarves of Faerun their own languages is as much in keeping with the 1e rule set with which he created the Realms as it is with Tolkien.
The dwarves of Faerun use runes, much like Tolkien's dwarves did. While there could be some inspiration there, the fact that dwarves live in mountains and work with stone makes runes kind of a natural choice. Further, Tolkien's dwarf runes and Greenwood's dwarf runes are quite different. This is in contrast to other games which lifted Tolkien's dwarf runes directly, such as the Ultima computer games. In terms of Faerunian dwarvish as an actual language, the other answer to this question demonstrates that it was never developed beyond a few words.
As for elvish, we see the same differences in Tolkien's written script and Greenwood's. While both alphabets are, I think most would agree, aesthetically pleasing in a graceful and artful way--in keeping with the character of elves--they are not really very similar, and Tolkien's written elvish includes numerous markings that go above the letters to assist in pronunciation, which Greenwood's alphabet does not have. Again, Faerunian elvish was never developed beyond a few words.
While there are similarities between the languages of Tolkien and Greenwood--the dwarvish use of runes and the elvish use of aesthetically pleasing characters--these similarities can be attributed more to the degree to which the elves and dwarves of D&D as a whole, not Faerun specifically, were inspired by Tolkien in their character and culture.