Hey firstname.lastname@example.org, I haven't tried this but I think you could do it with a combination of getting an RSS feed of a Twitter account and adding that to Canvas to post announcements. This is kind of the reverse of what you had asked for but it may accomplish what you're after.
Here's a site that talks about generating an RSS feed of a Twitter account:
And here is the guide on adding an RSS feed to Canvas announcements:
I mentioned that to the teacher but I think we're really looking for is whenever an announcement is built that there's an option to tweet it. Guessing that doesn't exist?
Sent from my iPhone
Twitter is also an LTI app that you can add to the course. You can add the Twitter widget to an announcement page or any page for that matter. Use the link that awilliams posted for instructions on creating the widget in your Twitter account.
It will look like the screen shot below.
it is worth noting that Canvas can push notifications for announcements, conversation messages, grades, and a whole bunch of other stuff to Twitter for students and instructors who choose to receive notifications via twitter instead of email or text message:
Perhaps that accomplishes the instructor's goal?