Can I post a link with a small description in the chat, about my MIT licenced GitHub repository?
Thanks for asking this in meta!
People in chat are there to chill, discuss about game dev and related topics (and random stuff when not talking about that), and for occasional drama. Some users are regulars and some pop in once in a while to ask a question or two.
Having someone pop in, post a link to a not really needed resource ("yeah, it's about gamedev, but I don't need a new engine now, I have mine already or I'm using Unity anyway") and leave immediately can be considered rude and spam-ish, specially if the user doing so is not seen around in chat often.
So, yeah, sure, you can post a link, but will it be useful to chat users? (They generally seek specific advice, so it's more like "pull" rather than "push".) And if a user finds it spam-ish and flags your message, and 5 10k-rep-users agree that it is spam, your message will be deleted and you'll receive a small chat suspension.
I would suggest that you hang in chat, wait until the discussion is appropriate and tell the specific users about your link when/if it could help them. If your link is about doing a very specific thing in Unity, there is no need to bother someone who's working with SDL or Unreal with it.
Another option would be to hang in chat for some time until users there get used to see you around, and post your project at that time. I'm sure people wouldn't mind once they get used to you.
Just don't annoy people :)
If it's relevant to the discussion going on, or if you want to start a discussion about it (as in asking for critique or opinions on it), then it's absolutely fine to post a link to your projects.