Applied for a faculty position and was notified three weeks ago that the Faculty Search Committee has placed my name on the short-list of applicants to be reviewed by all faculty in the department. I was also notified that if I am still interested in this position, I may be contacted within the next few weeks to be invited for an interview. However, three weeks have passed, I have not got any updated message. References were submitted by my referees at the time I submitted the application. Should I be more patient or just contact them by sending a follow-up email? Not sure what has happened. They lost interest to me or just act too slowly?
Congratulations on making the short list! There is no way to tell from the outside what the school is doing, but there is also little that you can do to change it at this point.
If the school's answer would change something for you in the short term - for example, if you have another job offer and want to know if the school is still considering you - then it would be very appropriate to send them an email.
If you are just emailing them to check on the status of your application, I would wait a few more weeks to see if you hear anything from them. This part of the application process is often a waiting game, unfortunately. On the other hand, I think that a very polite, short message would not be likely to harm your application, particularly if you let them know that you are still very interested in their department.
Unfortunately academic hiring processes are extremely amateurish (in comparison to corporate processes of most well established mid-size and large companies). You are being left in dark, because the university cares for itself (they are hiring!), not for you (you are waiting/hoping for a job). Being several times through the process, the simplest advice at this point is to wait. Alternatively, you can simply ask for the status. My experience is that once the process starts, you will receive an answer. Eventually. However waiting for the answer for several months is rather usual too.