How reasonable is it to ask about my application status to the Chair of the search committee? It has been more than 4 months I applied for a tenured-track faculty position in a research university. However, I have not heard back from anyone (e.g. representative of the Chair of the search committee or someone else) regarding my application status. Unfortunately, there is no online status-checker system available. I am even not sure whether my application has been rejected. In such a circumstance, can I place an inquiry to the search chair and ask him to let me know the status of my application?
It is completely reasonable to ask about the status of a job application at any time, let alone after so much time has passed.  Definitely feel free.

Since you have applied for a tenure-track job in the United States, I wanted to mention that my first thought was that it is overwhelmingly likely that the job search has concluded by the end of the academic year.  However, you say that you applied for the job "more than four months ago", which I guess means you applied in February or March.  That is an unusually late job application: most commonly applications are due sometime in the fall semester -- say, between September and December -- and the hiring process takes place either at the end of the fall semester or the beginning of the spring semester.  So if the application deadline was indeed in February or later, this is a somewhat unusual time (although such things certainly do happen occasionally) and it is not as overwhelmingly likely that the process has been completed.  Still the chances are not good: most American departments do not do tenure-track hiring over the summer, and once the new year starts there will be a flood of new candidates so they might as well start the process all over again.

For your future use, you might also want to look back carefully at the wording of the job application.  Some job applications have a "rolling deadline", and if you apply on certain automated academic job serves the application will be "accepted" in the formal sense for many months (usually; this depends upon the way the account is set up).  In most cases there is some text like "Candidates who apply before XX/YY/ZZZZ will receive full consideration."  You definitely want to get your application in by that date.  

Finally, I hope you know that with the current amount of job applications, many if not most departments make no effort whatsoever to contact candidates negatively.  It is also not in their best interest to do so: maybe another job opens up at a weird time or maybe the person who they thought would take the job cannot or does not for some reason.  If that happens they can go back and look at the applications that are still there and see if people are still interested and available.  Unfortunately the current academic climate is such that this not especially considerate policy has some strategic merit.