Twitter Sentiment Analysis: Detecting neutral tweets despite training on only Positive and Negative Classes
The quick (and not very satisfying) answer is "it depends" -- specifically it depends upon what your underlying conceptual model of human emotion is and how it manifests in verbal/written behavior. 

What is your characterization of neutrality in relation to positive and negative valence? Can documents be put on some sort of quantitative scale with neutral sandwiched between positive and negative? This position has linguistic support at least with simple phrases which express single valence states (the solution is {great > good > acceptable > poor > horrible}). 

As @dmb poster suggested, if this is your conceptual model, then you might reasonablly argue that neutral falls between pos/neg and all you have to do is determine the optimal boundaries/cutoff. Thus, you can assign neutral to a test case even if the classifier wasn't trained on neutral cases (though you do need some way of determining these cutoffs). 

But what about more complex cases? What happens when you move from the level of the phrasal unit to larger sentential and discourse level units? How would you rate the sentence "I love apples but hate bananas"? Do the positive and negative elements cancel each other out to create a neutral? You can easily see how muddied this gets when we start to talk about real human texts. In my opinion, I don't think it is particularly meaningful to talk about a document-level emotion score. Rather, I believe people express emotion/sentiment that is directed at an individual objects "I hate bananas" -> hate(subj, obj) that get combined into discourse-level constructions.

So, yes, I think you could reasonable defend the use of outputting a neutral category if only pos/neg are used to train... BUT you will need to justify your use of particular cutoffs as well as consider how to deal with longer documents that express multiple (conflicting) emotions. My preferred course of action would be to have people read and label tweets as pos, neg, neutral, and mixed (then ensure inter-coder reliability) to create my training data... let the algorithm do the hard work of finding the cutoff values.