Done this for quite a while and I love it. Real computer automation. This is an instructable to get your linux computer to access twiiter via the command line for both sending and receiving tweets. The great thing about this is it will work on a system without a gui. An older computer would be perfect for this project. In fact I run my tweeting computer without a monitor and access the tweeting computer via ssh. You can even receive tweets and turn them in to commands to be executed by the computer. That saves having to open a port to your network to access it. Gave you just enough to work with.
Another instructable, but you could add software kind of like the Dr. Eliza program to almost have a natural language interface. This could be an automated help center or even some kind of auto response chat line. You could go one step further and have the tweeting computer issue commands to other computers behind firewalls without having to open a port or human intervention all via a twitter account or accounts.
Note: Twidge is available for the Apple Macintosh and you could probably use almost the same batch files.. Microsoft Windows computers may be able to do the same thing with a program called "tweetc". You definitely you have to use different batch files though. (not a subject of this instructable).
Update: Set up your own twiiter like server: https://www.instructables.com/id/Statusnet-the-Twitter-clone-setup/