It would be tedious as hell, especially now that Twitter is all-Ajax-all-the-time, and it would be a moving target as they made changes, but could Twitter really do anything to stop it?
I think the answer is no, they can't stop you. And perhaps I'm being naïve, but it doesn't seem like it would be too terribly tedious if you used a scraping library, at least not for replacing basic API functions (i.e. getting a user's recent tweets).
Obviously this would be limited to public tweets (no private tweets, no tweeting on the user's behalf, and no DMs).
I don't think any of those limitations apply. If you can do something via a web browser, you can do it programmatically, depending on how much pain you're willing to endure.
Yes, it's called iframe busting. You can't force a page into an iframe that doesn't want to be there.
Let's just say I wish I had the time for such an endeavor, and I sincerely hope someone out there does.