Tolkien hated Narnia: the two dons may have shared the same love of unquestioning feudal power, with worlds of obedient plebs and inferior folk eager to bend at the knee to any passing superior white persons - even children; both their fantasy worlds and their Christianity assumes that rigid hierarchy of power - lord of lords, king of kings, prince of peace to be worshipped and adored. But Tolkien disliked Lewis's bully-pulpit.
As I recall this was not Tolkien's reason for disliking it at all; what he didn't like is the carelessness of the creation. Tolkien took artistic creation (sub-creation, as he called it) very seriously, thinking that the author should craft a consistent, coherent world with its own inner logic. Lewis piled a whole lot of things together, from talking beavers with sewing machines to Fauns to Father Christmas. This sentence from The New Yorker is a bit better:
Tolkien hated the Narnia books, despite Lewis’s avid sponsorship of Tolkien’s own mythology, because he hated to see an imagination constrained by the allegorical impulse.
It is true that Tolkien had a distaste for the 'allegorical impulse', particularly the tendency to read every story as an allegory. But, again, I can't recall that this fits with anything Tolkien actually says about the Narnia books; and one could just as easily say that Tolkien's tendency to dislike allegorical reading would have led him to be irritated by those who insist on reading the Narnia books only as an allegory -- it isn't necessary, and many people don't, for the very good reason that the books are not properly allegories at all, except in the sense that any book making extensive use of certain kinds of imagery will have allegorical tones, which can as a matter of art be explicitly harnessed. In that sense Tolkien himself does the same thing, and admits it: his descriptions of Galadriel are influenced by the imagery of the Virgin Mary, etc. It doesn't follow from these things that The Lord of the Rings is an allegory. (Eliot's Middlemarch taps into the imagery of earth and vegitation; it doesn't follow that Middlemarch is an allegory for the earth. Pullman's His Dark Materials series makes rather extensive use of Miltonic imagery, as interpreted by Blake. It doesn't follow that it's an allegory for a Blakean worldview, just that the imagery is supposed to have Blakean-Miltonic tones in acting on the reader.) Lewis denied (quite rightly) that the Narnia works are allegories in any proper sense. He was trying to write books that will strike readers the way MacDonald's books struck him, namely, those that 'baptized his imagination' in giving him an imaginative sense of Something More; but his mode of writing wasn't particularly allegorical -- he started with a few striking pictures in his head and wrote about them. The allegorical associations just followed from that. Further, the above description of Tolkien's complaint is very implausible; Perelandra, which is a re-telling of Paradise Lost, is even more easily read as allegory than Narnia, and Tolkien loved it. Tolkien's complaint was not about the content, but about the art: he thought the works weren't serious enough.
(As a side note, I find interesting some of these secular criticisms of the Christian-like content of the Narnia books, because they often seem to echo things Lewis attributed to his own atheist years, e.g., an appreciation of certain sorts of myth, but a revulsion toward the same kind of myth when it was put in Christian form. At least, I very much doubt they would make similar criticisms of a book about the Corn King or Odin on Yggdrasil. And what is it with all these people who seem to think that the lion is not a traditional symbol of Christ? The funniest point in these criticisms, though, is found in the first review I linked to above; she calls the books 'Republican', which seems a teeny-weeny bit of a stretch!)
