Twitter will normally respond to DMCA (Digital Millenium Copyright Act) requests that involve the "use of a copyrighted image as an profile photo, header photo, or background, allegations concerning the unauthorized use of a copyrighted image uploaded through our photo hosting service, or Tweets containing links to allegedly infringing materials."
Once a takedown request has been filed, Twitter notifies the user and gives them the ability to file a counter-notice to dispute the copyright claim. None of this has changed. What has changed is how the rest of Twitter will see these particular tweets:
"In an effort to be as transparent as possible regarding the removal or restriction of access to user-posted content, we clearly mark withheld Tweets and media to indicate to viewers when content has been withheld (examples below). We also send a copy of each DMCA notification and counter-notice that we process to Chilling Effects, where they are posted to a public-facing website (with your personal information removed)," says Twitter in its Help Center.
That is a pretty big shift from how Twitter used to handle removed tweets. In the past, these tweets would just vanish without a trace. Now, even though the content of the tweet will be removed, users will at least be able to see that there was in fact a tweet there at some point and it fell victim to a copyright claim.
Here's how a tweet (with an infringing link) will look when withdrawn:
And here's the look for withdrawn tweets featuring media:
Earlier this year, Twitter reported 4,410 DMCA takedown requests for 2011.