WARP CORE: In Beyond the Farthest Star the hatch to the warp core collapses on Scotty. It seems that the warp core is a common feature on starships from TMP onward, but not TOS.
Well, they actually called it the "engineering core," which probably meant the core section of the engineering complex. A better precedent might be "One of Our Planets is Missing," which showed a vast warp reactor with what seemed to be power transfer conduits pulsing with energy. Although that was supposed to be inside one of the nacelles.
And the engine shaft in TMP was called an intermix chamber. The term "warp core" is from TNG.
Let's see... later Trek shows made occasional references to "Edosian" plants and animals, a reference to Edos, Lt. Arex's homeworld (although the possessive/demonym used in the Alan Dean Foster novelizations was "Edoan"). And TVH featured felinoid aliens who were reportedly meant to be Caitians. It's been alleged that the tailed twins Kirk slept with in Star Trek Into Darkness
were also meant to be Caitians, even though they had no feline attributes other than tails.
More a practice than a concept, but TAS was the first incarnation of Trek to depict female security guards ("The Survivor," "The Lorelei Signal"). Meanwhile, "The Time Trap" was the first depiction of a sane, intelligent Orion woman, a precedent followed by ENT: "Bound" and the 2009 movie as well as a number of novels and comics.
"Lorelei" showed the transporter as a means to reverse artificial aging, a technique also used in TNG: "Unnatural Selection."
"The Slaver Weapon" was the first Trek story to mention stasis fields, although of course the concept came from Larry Niven's Known Space universe.