[EDIT: My basic thrust here is unchanged from my original Answer, but I have added quite a few critical and contextual references from past work that I have done in this sort of area. Despite having enjoyed Star Trek since its early UK broadcasts, when I started my academic research on science fiction I actively hoped largely to avoid using this show, precisely because of its naively-understood and disproportionate profile, both within fandom(s) and further afield, and both pro and con. Of course for that very reason its cultural gravity cannot be ignored, so I did become quite knowledgeable about the show itself, and about the absurdly idealistic legend that has grown up around it. So, I have done a lot of research, and I am happy to share some of the relevant bits here!]
[EDIT 2: I have just discovered someone's wonderful upload of the CD notes for Star Trek: The Original Series soundtrack collection, which makes no mention of the (pretty much spurious) lyrics, but provides brilliant background for the original composition and recording, placing those lyrics even more firmly in context. Lucky, cos the CD itself seems to be very hard to get.]
There are lyrics, technically speaking, but the simple term original is surprisingly misleading. That single, innocent little word opens-up a massive can of worms about where those lyrics really come from, their legal and ethical status, and even whether or not they are musically valid. With the evidence that we now have, it could be argued that anyone writing a respectful and relevant lyric would have more right to reward for it than Roddenberry, who (among many other questionable moneymaking practices) lumpishly took advantage of his position to effect a cynical but unstoppable smash-and-grab on the composer's royalties.
Those lyrics have been performed, but they never sound very good (I will get to that, below). Various attempted performances and first-hand accounts of the writing establish that the lyrics were not in fact designed to be performed in any case. I will provide some detail on all of this.
sfhq_sf has drawn our attention to this live rendition by Tenacious D, which relies strongly on (and still almost overpowers) Jack Black’s remarkable falsetto abilities.
Others have had a go, for example this fan, ‘Skybolt’, who prefaces his performance by saying:
I recently found out that the original Star Trek theme had random
lyrics written for it so the composer would lose half of his royalties
for all time. Way to go, Gene! Live long and profit!
Note that even while bracing for performance, as a fan, ‘Skybolt’ still describes the lyric as ‘random’. He audibly wrestles with the vocal range required to deliver this, veering between octaves (i.e. he is forced to drop dramatically in pitch, where Black can just about use falsetto.).
Tenacious D give another (apparently impromptu) rendition for the Nerdist podcast, once again making very clear the vocal challenge. Jack Black gives more detail this time:
These lyrics were written by Gene Roddenberry, and I contend that he
never intended for them to be used in the Star Trek opening credits.
But because he wrote them, and officially they are part of the song,
he got another cheque. He would also get his weekly cheque for being
the richest, you know, TV show creator... but he would also get
another greedy cheque for the song, which he really had no part...
He shouldn’t have taken half of that money.
What happened was this (and this is relevant to performance)...
Star Trek originally aired from 1966 to 1969. Its theme tune was composed and recorded by Alexander (‘Sandy’) Courage, presumably in 1964 or 1965 during production of the original pilot, and was as recognisable and distinctive as we all know. David Alexander, writer of the authorised (and therefore worshipful) biography Star Trek Creator: Authorised Biography of Gene Roddenberry (1994) blithely states (p217) that, ‘Gene settled on Alexander Courage to write the music and conferred with him on what he wanted.’ Courage himself, however, in an interview in 2000, says:
Bill [Wilbur] Hatch was the Head of Music at Desilu [Star Trek's
original studio, soon acquired by Gulf+Western, and merged with Paramount], and Star Trek was a Desilu production, so he said,
‘Listen, how would you like to write a theme for this thing, and we'll
see if Roddenberry likes it.’
Well, I did it in a week, between working at Fox, and, I don't know...
there must have been about half an hour of music at least in it.
And that was it: I composed it, orchestrated it, and conducted it.
[Interviewer asks whether it was recorded in one day.]
Oh, yes: it was done in one session! [Courage chuckles] A
morning, or an afternoon: sort of 10 to 1, or 2 to 5, or something
Alexander is not exactly lying, but he is briskly helping us to assume that Roddenberry was in hands-on control of the music from the outset, which would be (for Roddenberry’s image) a convenient frame for the way that events were to develop. The truth, however, seems to be that the music was commissioned, written and recorded very quickly, shoehorned into Courage’s busy schedule prior to Roddenberry being involved at all. Alexander’s account subtly implies a degree of involvement and ownership over the music that seems to be inaccurate (even impossible), and we will quickly see why he might try that.
In 1968 (between Star Trek’s second and final seasons) there appeared a book called The Making Of Star Trek, clearly (I’ll get back to this) credited to Stephen E Whitfield and Gene Roddenberry. On the unnumbered p7 (in my copy), before the book’s main substance, these lyrics suddenly appear, with no explanation at all, for the first time in public view. This publication is what guaranteed Roddenberry half of the royalties that would otherwise fully belong to Sandy Courage. Roddenberry had nothing whatever to do with the composition.
Furthermore, Roddenberry also had almost nothing to do with the book, either. Whitfield (later aka Poe, incidentally, e.g. as author of A Vision of the Future: Star Trek Voyager) first encountered Star Trek as a rep for the plastic kit company AMT (seeking to market a model of the Enterprise). He then got the idea of documenting the production, and Roddenberry agreed to that in exchange for co-writing credit (i.e., basically, half of the royalties).
On his StarTrekFactCheck site, Michael Kmet reports examining Whitfield’s final manuscript for The Making of Star Trek (now held at UCLA), and confirming that Roddenberry’s input to it was negligible. The (large!) manuscript contains a very few amendments by Roddenberry, reputedly executed in a single evening, and the final book does not deviate further from it. Ostensible co-author Roddenberry's input is clearly a great deal less (in terms of both effort and substance) than even a decent editor would have delivered. It does, however, irrelevantly contain those lyrics!
In their book Inside Star Trek: The Real Story (1996) Herb Solow (Executive in Charge of Production on Star Trek) and Bob Justman (Co-Producer) confirm that this co-authorship deal with Whitfield/Poe was made in 1966 (pp401–02), with Roddenberry saying, ‘I had to get some money somewhere. I’m sure not going to get it from the profits of Star Trek.’ Sandy Courage’s Star Trek theme music was already composed and recorded, and the book would not see the light of day for a couple of years David Hipple goes into this and other aspects of the Roddenberry legend in ‘The Accidental Apotheosis of Gene Roddenberry’, Chapter 2 (pp22–40) of Lincoln Geraghty’s book The Influence of Star Trek on Television, Film and Culture.
Many writers (including his senior colleagues Solow and Justman) document other instances of Roddenberry getting away with one unethical stunt after another, precisely because no-one could quite believe that he had the nerve to keep doing things like that: ‘The upshot was that everyone pretended not to know what had happened. So it continued to happen.’ (pp400–01) Setting up a personal business to sell Paramount property, and pretending legally that it wasn't his operation, was rather an impressive one. (Lincoln Enterprises was nominally his lover Majel Barrett's business, so it and its income could not be evaluated as his assets in the divorce proceedings that would inevitably follow this latest-of-many extramarital affairs coming to light.)
Another was actually organising the famous and supposedly spontaneous fan mail campaigns that legend enshrines as having saved Star Trek from cancellation, when the show did not in fact need saving at all. Those campaigns in fact embarrassed and inconvenienced both studio and network, and contributed strongly to Star Trek being sidelined in the schedules and then cancelled soon afterwards. (Mission: Impossible, which was commissioned by Desilu alongside Star Trek, ran on comfortably until 1973, and for a couple of years was a refuge for Leonard Nimoy, following Star Trek’s failure.)
As part of all this, it is not in the least credible that Roddenberry and Courage really collaborated on the music at all (Courage explains in an interview that Roddenberry interfered damagingly with the final mix, for sexually-motivated reasons!), or that the lyrics are in that sense ‘original’. They were slapped-on, after the fact, by the only person in a position to whip half of Courage’s proper royalties out from under his nose. On top of the way people observably struggle to perform these lyrics, there is evidence (beyond the very lumpy poetry) that they are badly-designed for delivery in the first place.
In the same book, Solow and Justman devote some space (pp178–85) to the ‘lyrics’ for Star Trek’s theme tune. In this very dense book they bother to include four pages of score, dated 1966, as if to prove beyond doubt this ethical offence, among others. Believe it or not, however, their overall account is not essentially anti-Roddenberry! They end up feeling quite sorry for a comprehensively marginalised has-slightly-been, still with unshakeable delusions of his own importance, kept on contract for the Star Trek films (but with no real power at all) by a reluctant studio with no professional faith or personal trust in him whatever, purely because it thinks that that might keep some ticket-buyers happy. (And, for balance, on the fervently pro-Roddenberry side we also still have the likes of Yvonne Fern's Inside the Mind of Gene Roddenberry, 1995, resolutely promoting the idea of a utopian humanist as she sits at his feet to absorb his dying wisdom.)
The explicit let's-debunk-Roddenberry torch is carried by Joel Engel's Gene Roddenberry: The Myth and the Man Behind Star Trek (1994; and there even the critical Engel's conscientious research is still taken-in by a few aspects of the sprawling Roddenberry legend). Even David Alexander's authorised hagiography is unable to sidestep all of Roddenberry's self-interested schemes. Alexander feels compelled to mention this royalty-splitting stunt, if only very briefly (pp235–36), giving the most space to a 1967 letter where 'Gene wrote to Courage in an attempt to straighten things out' (two or three years after the manoeuvre!). In fact the letter is not really conciliatory at all: in it, Roddenberry claims never to be unfair, and invites Courage to improve his own memory of how very reasonable the original arrangement was.
Engel uses exactly the same letter as the focus for pointing-out a couple of other things (pp106–07; and it is worth pointing-out that Alexander's book and Engel's were published in the same year: they cannot be arguing with each other in any substantial way, although [p44] Engel is aware of Alexander having sycophantically interviewed Roddenberry in 1991). It would be routine for a composer's contract (and Sandy Courage's was conventional) to include a note that should someone else contribute lyrics to a theme then royalties are split with the lyricist (i.e. collaborator, in effect). That would seem entirely fair in the case of Happy Days (let us imagine; I don't know the credits), where the lyric makes an obviously relevant contribution to broadcast. The point here is that Roddenberry used that convention simply to snaffle half of Courage's money (for what later became his outstandingly iconic piece of music)... by writing a poetically amateurish and borderline-unperformable 'lyric' for music that was never intended (by anyone, ever) to be broadcast or even simply performed with a vocal at all.
Engel then shows that Roddenberry performed much the same disappearing-trick over the royalties for Leonard Nimoy's album, Mr Spock's Music From Outer Space (1966). The point here is that while Roddenberry grumbled that he had invented Spock (and I do sympathise with conceivable distress at such a successful creation being freely hijacked for someone else's marketing), Roddenberry did not take on the record company (Dot) with an argument based on intellectual property. He just used an easy loophole that simply (if unfairly) entitled him to half of Nimoy's money, for Nimoy's work, if the record included music—which of course it did, even though it was an area that Roddenberry himself knew nothing about and had no professional involvement in.
Roddenberry's professional strategy is never motivated by principle: it is always the simplest route to securing cash, with as little effort as possible. There is considerable evidence in the documented history to suggest that one Leonard Maizlish (among other things the supposed 'owner' of Lincoln Enterprises), might have been behind Roddenberry's embrace of this kind of merciless financial opportunism. Everyone willing to comment appears to have regarded Maizlish as a predator, to be avoided.
Solow and Justman simply give an objective and detailed account of production, with impeccable sources. From that, and completely unlike the image projected by the legend that Alexander would like to support, the real Roddenberry that emerges is consistently and exclusively motivated by money and sex (for him, very much linked together).
On p178 they say this:
Roddenberry, who had never written a music lyric, chose to write one
for the Star Trek theme, a lyric that would never be sung or used in
any manner other than being printed on the published orchestral and
band arrangements. Its purpose was not an expression of creativity;
it was purely financial. Roddenberry, the lyricist, would now receive
fifty percent of all music royalties; and Courage, the composer, would
receive the remaining fifty percent.
On p185, Solow and Justman quote Sandy Courage as saying,
Roddenberry’s lyrics totally lacked musical practicality. He
made two very serious errors in writing the lyrics.
One, he changed the shape of the melody by adding extra beats, and two,
he used a closed vowel with a z-z-z-z-z sound on the highest notes,
something that gives great problems to singers.
So, in summary...
Yes, there are lyrics, if we can count Roddenberry cobbling-together a bit of a poor-quality filk using Courage’s existing music.
Yes, those lyrics have been performed, by courageous and/or energetic souls happy to attack the challenge. This would be a bit like trying to read the whole of The Eye Of Argon to an audience, with a straight face, while unicycling and drinking beer.
But... Even those who have performed this ‘piece’ want us to know about the ridiculousness of it, and the poor (and now late, 2008) bloke who did the actual work and had half of his money nicked was happy to explain in public (for anyone to check and challenge) why Roddenberry’s cash-in was musically amateurish. That, coupled with poetic superficiality, seems to explain why there appear to be no particularly good performances of this thing anywhere. With the lyric, this economically powerful composition... well... suddenly it isn't very good.
Says something interesting about the experience and professionalism of composers, anyway.