Deliberate mistake: At the very end, the Talosians send a final visual transmission of Vina and Christopher Pike, now whole and happy and reunited after 13 years, holding hands as they enter the Talosian elevator in the hillside. However, in this last shot, the elevator is still half-disintegrated, exactly as it was 13 years earlier when the Enterprise crew destroyed the hillside with a laser cannon. Within the context of "The Menagerie" storyline, this suggests that the Talosians never attempted to repair the elevator for 13 years (even though they continued using it). This incongruity is due to Gene Roddenberry cannibalizing his Star Trek pilot "The Cage," which contained zero footage of Jeffrey Hunter and Susan Oliver entering the intact elevator together (only the destroyed elevator). So, Roddenberry deliberately tried to "slip one by" the audience in this brief shot.
Suggested correction: There are reasons why the elevator would appear damaged. As the Talosians were in control of everything shown on the ship's viewer, the entire scene could be an illusion, or at least the elevator's condition may have been, with the Talosians choosing to allow the viewers to see the elevator in the same condition they last saw it. Just as likely, however, is that the Talosians truly never did reconstruct the elevator, as the whole point of their having a menagerie of other beings was an attempt to breed a race that could physically serve them, for their concentration on their mental powers had led to a complete inability and unwillingness to perform physical tasks (like repairing an elevator).
Still, as long as the Talosians are creating the illusion of Christopher Pike and Vina in their "restored" bodies, why not create an illusion of the elevator and hillside restored, as well? One big illusion of restoration, rather than a composite of dismal reality and happy-ending illusion? Again, to the point of my original post, the obvious incongruity is due to Roddenberry using the only happy-ending footage he possessed, that of Pike and Vina entering the half-obliterated elevator as they did at the end of "The Cage." Certainly, if Roddenberry only had the foresight to shoot Jeffrey Hunter and Susan Oliver entering the intact elevator, he would have used that footage instead. Any attempt to explain away the 13-year incongruity is mere wishful thinking.
This would qualify as a question, not a mistake. It is entirely plausible that the Talosians wouldn't bother to repair the elevator. It's also possible, as the previous correction points out, that the entire scene is an illusion. Remember, Captain Kirk sees Vina and Pike together on the planet literally moments after Spock wheels Pike out of the room. It's unlikely Pike had already been beamed down.
Deliberate mistake: Here, as in nearly every other ST episode featuring caves, caverns or mines, all the floors are perfectly flat and the lighting adequate-to-see-by or even brilliant. These configurations are easily found on soundstage sets - but not in nature. (00:16:30)
Deliberate mistake: When the Klingons post signs for their rules, they're printed in English. The Klingons have their own language, and people on Organia most certainly aren't English speakers. Even if the Universal Translator allows Kirk and Spock to hear in English, it wouldn't change the look of a sign.