How soon after the application deadline do search committees start looking at the applications? I notice that a lot of academic faculty positions have deadlines that are on a weekend, or during a holiday. It seems reasonable that an application will not be read until the holiday is over.

How soon after the application deadline do search committees usually start looking at the application?
The actual timeline for filling a faculty position is often almost entirely decoupled from the stated deadline.  The stated deadline is often set through interaction with an HR department or other formal process, and may or may not reflect the hiring committee's actual expectations of when they will get the applications in which they are most interested.

In the case of applicants who are likely to be seriously considered, the committee will often have effectively begun considering their application before they ever submitted it, based on interactions at conferences and other forms of networking.  Other applicants are likely to be triaged as they come in, with likely prospects given serious consideration quickly and unlikely prospects.
In the end, however, every faculty has its own process and expectations, which may vary widely: in some cases, they may be closely tied to the deadline, and in other cases they may have little or nothing to do with.