I can't think of specific character scenes but there are lots of missable scenes,
When you are fleeing Figaro castle in the beginning of the game, you have until you get to the South Figaro Cave to see the scene where Locke and Edgar freak out about Terra's Magic. If memory serves me correct that is the requirement not simply using it during the battle with the Magitech armors but I could be wrong...
Taking Edgar to the small house north of South Figaro, have him look around and he'll comment on how he recognizes the flowers and dishes which leads him to make a comment about his brother Sabin.
You get two different scenes depending on whether you agree to help the returners or not but you actually get more dialogue for refusing.
You get different dialogue from Celes depending on what Locke is wearing when he meets Celes. The best imo, is the soldier uniform cause you get a Star Wars reference.
Most people I know miss out on doing the letter sidequest in Mobliz, you can actually help the poor soldier send letters to his girl and get more info about what's going on in the Empire, you also get a decent item.
There are several scenes in the cave where Gau's treasure is located a player can miss if they don't explore every square inch of the cave. Most of it is funny including Gau scaring Cyan costing the party a bit of gil.
Also, there is an amusing scene with Cyan if you talk to the show girl in Nikeahe
There's the Coin Toss scene if you bring the Figaro Brothers to Figaro Castle and spend the night.
Shadows dreams which can be seen from staying in the inns on the western continent.
There's the NPC that tells Kefka's backstory in Vector but he's located in a place most people don't bother looking. He's in the bar which is the building located just before the stairs leading to the Imperial palace. Its not a scene per se but I'm surprised how many people miss talking to this NPC.
There is a scene with Setzer that some people miss. After the espers level Vector and you are sent on the quest to find them in Thamasa; if you go back to the airship, you'll find Setzer and Cid repairing the ship. Cid is being a nuisance to Setzer but Setzer ends up talking about Darly and foreshadows the Falcon.
There are several scenes in Thamasa you can miss if you don't thoroughly explore the town before talking to Strago. Most involve the citizens trying (very badly) to hide the fact they can use magic.
Oddly enough, after the events of Thamasa but before you go to the Floating Continent, if you return to Thamasa with Strago and Relm in your party, you'll notice a new NPC. It's Gung-Ho who will berate Strago for giving up on the search for the Hidon, foreshadowing the quest in WoR.
You can miss Celes' suicide attempt by saving Cid (yes you can save him and no it is not worth it).
Obviously enough, everything pertaining to Shadow is changed depending on whether you waited for him at the Floating Continent.
There is the Gau scene you mentioned, and I suggest recruiting everyone before watching it cause its awesome seeing the whole party in the shop.
Taking Terra with you into Cyan's dream world allows you to get access to her full list of Magitech spells when you are riding the armor.
The scene with Cyan talking to the girl who sends letters to the wounded soldier that BMN mentioned.
Then ending is altered if you don't recruit certain characters, Mostly if you don't grab Sabin, Locke, and Strago.
That's all I can think of though I'm sure you knew most of them.