No, they do not charge you extra… unless you are roaming. However, roaming nowadays no longer means just being in a different town or state like it used to. Most plans these days only charge roaming fees outside of the US.
But even then, it only matters if you actually use the thing. Most smartphones will periodically transfer little blips of data (Facebook updates and such) but an old dumbphone only counts if you are making/receiving a call or sending/receiving text messages.
It sounds to me like either somebody is so technologically inept that I am surprised that they even know how to turn their phone on/off, is still stuck in the ‘90s, or (more likely) they prefer to remain unreachable.