Per the title, people who appear in public (or on TV), that are always “in character”.
The first one that comes to mind is Dan Whitney, aka Larry the Cable Guy. Every time I see him on TV it is always as Larry. Haven’t seen him as Dan except once in an old YouTube clip.
The second one is Gilbert Gottfried, who is always, always speaking in that raspy voice. (I know there is a YouTube clip of him supposedly using his “normal” speaking voice.) Bobcat Goldthwait is another “strange voice” guy that comes to mind.
Dolly Parton. I’ve seen interviews in which she says that she won’t answer the doorbell without full hair and makeup. Only her husband knows what she looks like without it.
Bob doesn’t use his “Bobcat” persona that often anymore. I’ve seen him in quite a few appearances (TV, movies, and Wait, Wait, Don’t Tell Me) where he is “Bob”, not “Bobcat”
Colbert has done many interviews out of character, talking about the separation between him and “Steven Colbert”. The character will pretty much have to end after tonight if he’s going to be a successful host on late night.
Old time wrestlers were loathe to break kayfabe - the illusion that they were really as presented in the ring. It wasn’t until relatively recently that everyone acknowledged the act.
Though there were a few exceptions, Harpo Marx generally stayed in character and did not speak in public. He could get away with it partly because the character wore a wig; without it, he wasn’t easy to recognize.
I suspect Paul Reuben’s crash after that Sarasota movie theatre incident wouldn’t have been so terrible if he had appeared as himself on talk shows (like Letterman) rather than as Pee Wee Herman.
Clips of Teller (of Penn & Teller) speaking are not particularly hard to come by on YouTube, but in general he seems to keep to his schtick when appearing with Penn.
These days, kayfabe is pretty much dead dead dead, but there are one or two who still make an effort at keeping kayfabe outside the ring. The Undertaker was well known for protecting his character, though he’s relaxed quite a bit this year. These days the guy mostly known for staying in character outside the ring is Bray Wyatt–he does local news interviews and the like pretty much in character, though it’s more of a blend of in and out of character.
Only while performing as Penn and Teller. Both Penn and Teller greet every fan after each show. Teller is quite chatty. He has also been in a few acting roles in which he speaks. For some reason this forgettable movie pops into my head. He speaks in that one with a southern accent IIRC.