Appropriate process for following up on university job posting
Note: I'm assuming this is for a regular staff or research staff (lab tech/manager but not post-doc) position. I'm also going to assume you're in the USA.

We just ran a search for a staff position in my program at my large research university. The union rules for my university specify that we can must first open the search for internal candidates before allowing externals to apply. If there is an internal candidate, they get priority before any externals can be considered.


Internal candidate: Someone already with a position in the university staffing pools looking to transfer jobs or someone in the university layoff pool
External candidate: Someone who does not have a job with the university and was not laid off from the university in the last 180 (or so, union rules are byzantine) days


Although we had some good external candidates, we essentially could not consider them as we also had good internal candidates. Contacting the lab manager or program head (in my case) about the search via non-HR approved means would not have been productive. 

You might as well try (what are they going to do, fire you?) but be prepared to be turned down or have your e-mail ignored as university hiring tends to be done strictly by the books in most American universities.