Did Richard A. Knaak get the idea for Warcraft from J. R. R. Tolkien?

Warcraft borrows far more heavily from Games Workshop than directly from Tolkien. Green Orcs are not Tolkien; they're pretty much Warhammer, tho'. Warhammer's success resulted in the green orc meme (and "ork" spelling) escaping into the wild.

Even the nature of the game is far more Warhammer than it is Tolkien. Tolkien shows a few brave men turn the tide; warhammer is unending war, and heroes make no lasting difference. Warcraft is far closer to that.

Further, anything licensed by Tolkien's estate clearly bears the license mentioned; you'd see either a Tolkien Ent., Christopher Tolkien, or Saul Zaentz license announcement on every boot if it had been.

It can be said fairly that, without Tolkien and LOTR, we probably would not have seen anything with the particular tropes of Warcraft, but it's only by filtration through multiple generations of games that the distinctive elements come to be present in Warcraft.

For completeness: Warhammer attributes (in early editions) much of its material to parody of both AD&D and Tolkien. It's a remove or two from Tolkien. Inspired by AD&D, it parodied it, and integrated some ideas direct from Tolkien in parody, others in tribute, and many more come from the AD&D warpings of Tolkien and other fantasy writers' takes on Tolkien.

Without authorial statements to the contrary, it's rather easy to point out that LOTR is not the most likely candidate for the primary source. It's rather more difficult to say that Warhammer is specifically a source point, but it's not hard to point out that it's more likely, by preponderance of the evidence, to be the divergence point than LOTR itself.

We can clearly rule out a license, since no text indicative of a license is present, and both Christopher Tolkien and Saul Zaentz are incredibly nasty about requiring appropriate license notices on the outside of anything licensed. They do this to prevent the various IP's from falling into public domain.

aramis hit the nail on the head, in his answer - Warcraft is inspired by Warhammer. Kotaku supports this:

Allen Adham hoped to obtain a license to the Warhammer universe to try to increase sales by brand recognition. Warhammer was a huge inspiration for the art-style of Warcraft, but a combination of factors, including a lack of traction on business terms and a fervent desire on the part of virtually everyone else on the development team (myself included) to control our own universe nixed any potential for a deal.

I also want to point out that Richard A. Knaak did not "get the idea" for Warcraft, except from Blizzard. According to BlizzPlanet:

They approached him because of his Dragonlance books. Some of the heads of Blizzard had grown up on his Dragonlance works and asked him to write for both Warcraft and Diablo.

According to Knaak's own website:

It was his work and creativity that caught the eye of Blizzards, and they made a deal with him to write the lore for Warcraft and Diablo.

and

It is interesting to point out that Richard Knaak played Warcraft II and a little bit of WoW in order to get acquainted with the lore.

See also this related answer on the similarities between Warhammer 40K and Blizzard's Starcraft.

On top of what everybody said: Tolkien and his estate do not have the copyrights over myth and folklore. Anyone can use elves, orcs etc. on their own works (as long as you don't brazenly and completely copy others' designs). Warcraft's fantasy races such as orcs, elves, goblins, orcs etc. are in the public domain and Blizzard never needed to ask anyone permission to use those.