I would probably start by mocking the specific parts of the API you need for your application. In fact, this may actually force you to come up with a cleaner design for your app, because it more or less requires you to think about your application in terms of "what" it should do rather than "how" it should do it.
For example, if you are using the Twitter Search API, your application most likely should not care whether or not you are using the JSON or the Atom format option. The ability to search Twitter using a given query and get results back represents the functionality you want, so you should mock the API at that level of abstraction. The output format is just an implementation detail.
By mocking the API in terms of functionality instead of in terms of low-level implementation details, you can ensure that the application does what you expect it to do, before you actually connect to Twitter for real. At that point, you've already verified that the app works as intended, so the only thing left is to write the code to make the REST requests and parse the responses, which should be fairly straightforward, so you probably won't end up hitting Twitter with a lot of junk data at that point.