Yes, the linked project does say they have a demo server at llm-over-dns.duyet.net. (I didn't bother to check whether it's still working.)

Edit: This particular demo server doesn't work. There's another LLM over DNS post from a year ago https://news.ycombinator.com/item?id=44813298 where the server seems to answer some queries but not others.

Edit 2: Actually the server in https://news.ycombinator.com/item?id=44813298 does work with queries like "what is the capital of france", I was querying with special characters like "what's" or "1+1". So yes there are people opening hosting these on the Internet.