This came up in discussion again, so I might as well write it down now.
Tolkien made a statement that has often been taken as offering his imprimatur to those who wish to adapt and recast his works.
In 1951, in a letter to Milton Waldman of Collins, a publisher he was hoping would issue his work, he wrote of his intent in creating it, "to make a body of more or less connected legend, ranging from the large and cosmogonic, to the level of romantic fairy-story ... which I could dedicate simply to: to England, to my country." He explained that "I would draw some of the great tales in fullness, and have many only placed in the scheme, and sketched." And then comes the key sentence:
"The cycles should be linked to a majestic whole, and yet leave scope for other minds and hands, wielding paint and music and drama."
All sorts of people have volunteered to be those "other minds and hands." They've even erected it as a motto. And they take this as his authorization.
There are several points which this interpretation leaves out.
1) Tolkien's description is of his former intent. He's long since given it up, disavowed it. The opening words of the paragraph, just before the "to make a body" I initially quoted, are these: "Do not laugh! But once upon a time (my crest has long since fallen) I had a mind ..." And just after the key sentence he puts one word: "Absurd."
He's both boasting of and apologizing for his project at the same time, a peculiar approach. It's evident that in his heart he still wants to do it. But that's not what he writes: instead, he is not endorsing the description he is giving of his former intent.
2) His idea of what the "other minds and hands" would do does not include literary storytelling. It's other arts only. "Wielding paint and music and drama," he says, arts he did not aspire to. In writing to visual artists and composers inspired by his works, Tolkien consistently hopes for a work "akin to my own inspiration" (letter to Carey Blyton, 1964), but is not offended if it is not. (Unless it was in an edition of his book.) But he was infuriated by proposed sequels. "I suppose .... that there is no legal obstacle to this young ass publishing his sequel, if he could find any publisher, either respectable or disreputable, who would accept such tripe" (letter to Joy Hill, 1966). Tolkien is not endorsing fan fiction.
What about dramatic adaptations, though? It's evident, especially if you read Tolkien's essay "On Fairy-stories" (originally written in 1938/9 and published in 1947) that he considered drama an art separate from literature. But he changed his mind.
3) This letter was written in 1951. That's before The Lord of the Rings was published, after which Tolkien belatedly discovered through some painfully inept attempts at dramatization of The Lord of the Rings how naïve he had been in his description of planning to lay his legendarium out in the public domain in the Waldman letter.
Read the preface to the second edition of The Lord of the Rings, which takes the perspective of a literary artist claiming ownership of and responsibility for his own writing. Read also the numerous letters expressing his dismay at early attempts at dramatization of the book in his letters for 1955-58.