Looks like Xbox 360 users under the age of 18 cannot use Twitter on Xbox 360.
Apparently, this goes for the whole Dashboard update that includes Last.Fm and Facebook as well, until parental controls allow it.
Here's the official word from the Major himself:
As some of you might be aware, those under the age of 18 are not able to access Twitter, Facebook and Last.fm in our public preview. I want to you let you know that this will remain the case when it launches to the world soon. We made this decision because as it stands now, parents aren’t able to use Family Settings to customize which of these applications their children can access. In order to offer an age-appropriate environment for everyone, we decided to turn off these applications for those Gold members under the age of 18.
As expected, this has been a hot topic with the preview users. We want to make sure everyone was aware, however, that the development team is working on an update that gives parents the choice of which social applications their children can access. This means that Xbox LIVE members between the ages of 13 and 17 will be able to use Facebook, Twitter and Last.fm as long as their parents approve. The team hope to roll this out several weeks after launch and I will update you on the exact date as it gets closer.