if a school is accepting applications for faculty through September 15, when does the job start? If a 4 year liberal arts college is doing a search for a faculty position through September 15, when does the job start? The start date is not mentioned. 

The position is for a new interdisciplinary degree program that appears to have been only been granting degrees for one year. It is teaching rather than research focused. I assume the interviews for prospective candidates last through the end of the year. Is it typical that this position is starting the following fall term, or would the hire start teaching in the spring?  Or is it institution dependent and I should just try to find out from the school?
Only the hiring institution knows for sure, so you would have to ask them.

However, the start date is almost always listed in the job posting, so to avoid looking silly, be sure to read the posting carefully, as well as any  accompanying information that is included or linked.  If you don't find it, then contact the hiring institution (the posting probably contains contact info for a responsible person).