Mozilla's pioneering Do Not Track initiative, which has since been co-opted into an industry standard, not yet finished, paved the way for a different approach to privacy, one in which users had at least some say and control.
Do Not Track is by no means perfect, but it's a big step forward. One of the big problems with this approach is that it depends on advertisers and third-parties adhering and respecting it.
The Do Not Track header simply informs a website that the user does not want his or her actions to be tracked. The advertisers may do as they please though. This is why it's important that advertisers and websites voluntary start adopting and respecting the setting.
That said, Twitter is not an advertiser in the traditional sense and not a large one in any case. It relies on Promoted Tweets, Promoted Accounts and so on, but these are not the type of ads you'd see on the web.
What's more, Twitter doesn't allow any third-party advertiser to run ads on the site, everything goes through its own team and its own tools, even if it has started allowing smaller companies to promote themselves on the site.
But Twitter does have a lot of info on its users, which it will not be sharing with others or even keeping for itself. While there are no details on what exactly respecting Do Not Track means for Twitter, it can be assumed that it means it will not be using the data it has on users in advertising. Hopefully, more advertisers will adopt the same stance without them having to be obligated by the FTC or via other means.