I am just wondering if anyone else is experiencing this problem with some members randomly getting a "Banned" user screen on the forum when using their mobile device?

Logging out then in again appears to fix this bug.

Given that it's happening on mobile devices, and IP Addresses change rapidly on mobile networks, it could be happening because they're using an IP Address which you've banned in the system. But you'd need the full message, with any error code, to diagnose further. (i.e. if the error number is #2000 that's an IP Address ban.)

If that does not narrow down where you can look, please file a ticket to Support for further assistance.