When the NodeJS app is closed, the connection will be terminated. After that point it will have no impact on your system or your network.
Chat on the Twitch site itself doesn’t use tmi.js, as that’s a 3rd party module, but under the hood it’s doing exactly the same thing and establishing the same sort of connection to the same places. In fact if anything chat on the Twitch site is doing even more as it also establishes connections to others services to enhance chat, where your app is just a single connection to chat.