A connection to Twitch Chat will have a negligible impact on your network, and wont do anything to slow it down or make it unstable.
A chat connection is very simple, low bandwidth, and creating a connection from a bot you wrote is no different than going to twitch.tv and using chat there, so even if there was a network issue you would be experiencing it each and every time you visit a Twitch page that has chat.
I would suggest you contact your ISP if you are having internet connection issues.