Why iTunes music can not play without Wi-Fi?
I traced the problem down to a TLS handshake and a total of 63 packets between the host and 23.211.148.217, which is init.iTunes.apple.com - that is executed when the current version of iTunes is started. Once this handshake completes, all is fine. But you must tunnel traffic to this host on startup, otherwise you get the error with each and every song change.

You can play music without wifi if you have an internet connection when you start iTunes. So, connect your laptop to the internet and do not turn it off/close iTunes, just put it to sleep. Then, the error will not appear. Not a fix, but it works.