Ryujin
Legend
wow no one said : this is a bad idea?
I will be the first: This is a bad idea.
As either a player or DM I would find this annoying, and a hindrance to RP and party unity. Few characters I have played would be friends with someone who would not talk to them. And as others have mentioned it will distance you from a big part of the game.
The telepathy/talking beast/NPC servant idea would fix most of the problems, but would inevitably get killed. A familiar might work better as you expect them to die, and bringing them back is only a short rest away.
If you must play this special snowflake, try it as a guest appearance for 2-3 sessions, and have a second, more party friendly character ready to go.
If it's role playing, and doesn't really put out the rest of the party, then there's no such thing as 'bad.' I would argue that what I did with a character created far more potential issues, than simply being a mute. My character, a Warlock, went everywhere while wearing a Noh mask.
Well let's see..... Gets power from some otherworldly entity and won't show his face? Instant distrust. Before our group became local rock stars, NPCs would ask me if I was some sort of criminal. Having a high CHA and being able to generate truly ridiculous Diplomacy, and Bluff numbers tended to eventually smooth things out, but we always started in the hole.
Also didn't help that the guy, who was automatically distrusted, was the group's 'faceman.'