I asked Mew to IP ban me to test this, so Mew IP banned me.
With a HTTP proxy I could still access the chat, including me posting stuff and seeing other posts + seeing others users online. This probably also means asking questions and stuff works as well.
I also tried using an anonymity network, and that worked as well.
It's pretty annoying when a user gets banned (IP banned as well) then just finding a proxy and coming back with a dupe :/