I applied for a faculty position. One month later they asked my references for reference letters. I have not heard from them for a few weeks. I have not been asked for a phone interview yet. Is it appropriate to touch base?
I am the chair of my department's faculty recruiting committee. Yes, you should definitely feel free to contact the search chair to ask about the status of your application.
My department started accepting applications last October for faculty positions beginning in Fall 2016. We automatically solicit reference letters for all assistant professor applicants, usually within a few days of the application. We have invited almost all of our interview candidates at this point, but we may still invite more, and in principle, we can even continue accepting new applications. Except for interview invitations, we normally do not contact applicants until the search formally closes, which will probably happen in mid-April.
It's probably safe to assume that if you haven't heard from a department (at least in the US) by now, they're probably not interested, but there are certainly exceptions. My own job search is an example. My current department invited me for an interview in late April, three weeks after sending me a rejection letter. ("We changed our mind; we'd like to interview you.") The framed rejection letter is hanging on the wall in my office.
I don't think it's unreasonable to touch base, but let me be a devil's advocate: my rule of thumb is that you contact people about the status of applications when you have new information for them; the best example would be an offer somewhere else, but that isn't the only possibility. If you don't have new information for them, what purpose is going to be served by contacting them? What could they tell you that will change how you're going to live your life?
Given that they haven't contacted you directly (I'm assuming since you don't mention it), you should probably just assume you are not in serious contention for this job. Maybe you will be at some point later in the process, but if that does happen, it can be a pleasant surprise.
You can certainly contact the department (or whoever from the search committee who contacted you asking for recommendation letters) and politely ask where the search process is.
For the search process that I'm currently involved in, you'd be told "We've invited candidates for on-campus interviews, but you weren't on that list." You might ask why others who weren't invited for an on-campus interview haven't been notified. One answer is that we might (in case someone declines to interview or people turn down our offers) go back to our short list and invite someone else. Another reason is that the human resources office is ultimately responsible for sending out "We had lots of very well qualified candidates, we're sorry that we didn't hire you this time." messages, and they'll wait to do this until after we've actually hired our candidates.
At a certain point it's likely to become public who will be interviewing on campus. If the department has a colloquium calendar, you can check to see if any presentations by faculty candidates have been announced. The field specific "X jobs wiki" pages often have useful information (although the rumors can also be incorrect.)
In my experience, it is not unusual for universities who are interested in you to wait 3-4 months or more before contacting you (or to never contact you if they are not interested in you). It's perfectly fine to touch base politely. Chances are they were planning for a distant start date and haven't yet put together the search committee, started reviewing applications, shortlisting, etc.
In what country did you apply for the position? In Germany it can take months between signs of life from search committees. You are not out until your application is sent back to you, I've seen this take multiple years. It could be that the people on the first shortlist all turn down the offer, so the search continues with the same group of applicants.