Even the first season had moments of levity within some of the most serious episodes; this isn't strictly a matter of changes between seasons 2 and 3. For example: Spock's various statements of what the odds against them were (and Kirk's reactions to them) in "Errand of Mercy"; the Chinese rice-picker story in "City on the Edge of Forever." And the less heavy episodes were full of moments that could raise a smile: "I'm going to have a son" in "Tomorrow is Yesterday"; "I'm about to start devoting my whole life to it" in "Mudd's Women"; etc. (I'm leaving out "Shore Leave" and "Squire of Gothos" as they were often overtly comic.) There is, as far as I can remember, no such moment in any season 3 episode, assuming again that "Spock's Brain" is unintentionally funny.
Well, there were moments of humor, but no out and out comedies. These bits were usually at the end of the episode, where you’d see them exchanging some light comments. Some examples:
: “I never should have reconnected his mouth.”
And the resulting yuks with music helping out.
The Enterprise Incident
: Kirk not wanting to have pointed ears, making everyone but Spock smile.
Whom Gods Destroy
: Kirk’s comment of “Mister Spock. Letting yourself be hit on the head, and I presume you let yourself be hit on the head, is not exactly a method King Solomon would have approved,”
wasn’t all that funny, but it was meant to be light and airy.
The Lights of Zetar
: “Well, this is an Enterprise first. Doctor McCoy, Mister Spock and Engineer Scott find themselves in complete agreement. Can I stand the strain?”
Otherwise, I agree, there are few to no real “humorous schemes” during a tough moment, like in COTEOF and so on. There was a real effort to keep the show very grim. I think that actually helped during the more weird episodes, much like the approach helped the Irwin Allen series, like Voyage and Time Tunnel. If the characters aren't winking at us, it's easier (for me) to take it seriously.