How to block "Twitter’s Promoted Tweets" on iOS
Twitter is injecting the promoted tweets into the stream. It's only a matter of time until they lean on every third-party app provider to stop filtering out the promoted tweets as this is how they're going to pay for the service.

In the short-term you can use a different client but in the long term all signs point to you being forced to see promoted tweets promoted tweets or leave Twitter.

Even in fall of 2013, most major third party twitter apps aren't showing promoted tweets. I use twitterrific and don't see promoted content in my streams.