I recall that in the beginning of Star Trek: The Motion Picture, McCoy beams aboard the ship with a fabulous beard. The next time we see him though, he's shaved it off. That got me to thinking that I couldn't remember any instance of a Starfleet Officer from the 23rd Century with facial hair. The 24th century, yes (Riker is the first that springs to mind), but not the 23rd century (TOS era). So, my question is: do we ever see a 23rd Century Starfleet Officer with facial hair?. Note, I'm interested in intentional facial hair i.e. I won't accept the occasional 5 O'Clock shadow which has grown simply because they didn't get the chance to shave - I'm looking for facial hair which was clearly intended.
We have Captain Chandra from TOS: Court Martial
And in the same vein as @N_Soong's answer, we do see two Starfleet Officers in the 23rd Century (courtesy of the episode DS9: Trials and Tribble-ations) sporting facial hair; Captain Sisko and Lt Cmdr Worf.
Obviously Worf is a bit of a push (he's in disguise) but Sisko meets Kirk and doesn't get shouted at for being out of uniform for having a beard, suggesting that regulations merely require a beard to be tidy, rather than entirely absent
We also see Lieutenant Kyle (who seems to get more hair as he rises in rank - see below) as well as Captain O'Shea wearing a full beard in ST: The Animated Series.
Lieutenant Scott sports some fine facial fuzz in ST:TMP
Captain Terrell and Commander Kyle in ST:II
Captain Styles and an unnamed Engineering Crewman in ST:III
An unnamed Efrosian helmsman and two unnamed Caitan Admirals in ST:IV
Yeoman Burke, Colonel West, Lt Cmrd Valtane, Lt Cmdr Lojur, an unnamed Galley Officer and two unnamed Starfleet Admirals in ST:VI
And here we can see various unnamed Starfleet Officer cadets in ST:TMP
Captain Krasnovsky (Court Martial - TOS S1Ep14)
played by Bart Conrad
Fleet Admiral Harry Morrow
Per Memory Alpha
Harry Morrow was a Starfleet Fleet Admiral who served as Commander, Starfleet in 2285.
Admiral Morrow was portrayed by Robert Hooks in Star Trek III - The Search for Spock.
If we count fur as hair
Another Caitian (thanks to Richard)
Unnamed Senior Officer (top left)
I wrote this question and only now have I remembered one: Captain Morgan Bateson of the USS Bozeman who we see in TNG 'Cause and Effect'
Although he technically is a 23rd Century Starfleet Officer, in the spirit of the original question, this would be overridden by a TOS- or TMP-era instance.
Who can forget Spock from the mirror universe...