Oh my goodness, I'm certain I forgot half of the concepts, but I'll give it a try.
has already mentioned the "Engineering Core". Parts of the same animation cel were used in "Practical Joker" to show the entry to the "Computer Core". "Engineering Core" was at least mentioned in two different TAS episodes.
Then we have, of course, the holodeck
from "Practical Joker".
TAS showed first that the "mouth" of the Klingon Battlecruiser
could fire energy projectiles and I think the final design of the TMP intermix shaft
was - as strange as this may sound - actually inspired by Klingon technology
seen in "More Tribbles, More Troubles".