Does obtaining citizenship somewhere help in applying for a faculty position there?
In general, employers in the US are allowed to discriminate between applicants who are already eligible to work in the US (by virtue of citizenship, permanent residency or some special immigration status) and applicants who are not currently eligible to work in the US (who would have to be sponsored for an employment based visa.)  It is not permissible to discriminate between applicants who are US citizens and those that are simply permanent residents or otherwise have work authorization.  

Sponsoring an applicant for an H-1B or other visa can be very time consuming and costly.  Furthermore, approval of such a visa application (and even more so for a green-card application) requires the employer to argue that no qualified citizen or permanent resident applicants were available.  There are potentially substantial penalties for not following these rules.  

If you're already eligible to work in the US, and you are otherwise well qualified for the position, then you probably will be more likely to get an interview than someone equally well qualified who does not have this status.  

From my experience of running search committees for a math department, I can tell you that (1) The widely used AMS application form includes a check box for applicants to indicate their status and (2) interviewing qualified applicants who are eligible to work in the US is an important consideration and (3) the vast majority (95+%) of our applicants would require visa sponsorship.