Question: I'm trying to predict the sentiment of the next tweet posted by a twitter user. Right now I have the following steps (step 1 and 2 are already implemented in python):

1. Learn how to classify a tweet as postive (1), neutral (0), or negative (-1). I use a naive bayes classifier for this and it works pretty well.
2. Classify existing tweets from a user. This results in a series of numbers like this: [0, 1, -1, -1, -1, 0, 1, 1, ..] There's also information about the posting time.

Would it be possible to predict the sentiment (1, 0 or -1) for the next tweet?

What algorithm could I use for this? 

I don't know how this one works yet, but are hidden markov models suitable or some kind of regression?

Answer: 
I think one appealing way to think about this is in terms prior and likelihood for sentiment. Naive Bayes is a likelihood model (how probable am I to see this exact tweet, given that it's positive?). You're asking about the prior probability of a the next tweet being positive, given that you've observed a certain sequence of sentiments so far. There are a few ways you could do this:

* The most naive way is the fraction of tweets the user has uttered which are positive is the probability that the next one will be positive
* However, this ignores recency. You could come up with a transition-based model: from each possible previous state, there's a probability of the next tweet being positive, negative or neutral. Thus you have a 3x3 transition matrix, and the conditional probability of the next tweet being positive given the last one was positive is the transition probability pos->pos. This can be estimated from counts, and is a Markovian process (previous state is all that matters, basically).
* You can get more and more complex with these transition models, for example the current 'state' could be the sentiments of the last two, or indeed last-n, tweets, meaning you get more specific predictions at the expense of more and more parameters in the model. You can overcome this with smoothing schemes, parameter tying etc. etc.

As a final point, I think @Anony-Mousse's point about the prior being weak evidence is going to be true: really, whatever your prior tells you, I think this is going to be dominated by the likelihood function (what's actually in the tweet in question). If you get to see the tweet as well, consider a CRF as @Neil McGuigan suggests.
On the machine learning side, you could consider sequential associations:

<http://web.mit.edu/rudin/www/RudinEtAlCOLT11.pdf>

This site has some java libraries:

<http://www.philippe-fournier-viger.com/spmf/>

A Hidden Markov Model should also work. An HMM is a special case of a Conditional Random Field, which lets you look at other factors such as say weather or news events. 

I wonder if the next tweet of a person is also affected by the current tweets of a) everyone b) or those that they follow