Utah, and Salt Lake City, was a bigger film draw in the 1980s and 90s, especially for major blockbusters. I've noticed, though, that most the movies filmed here are either Disney made-for-TV stuff or really low budget movies no one will ever watch anyway.
I'm not sure what's changed the past decade or so that's led to less bigger movies being filmed here, but you can definitely tell they've tapered off a great deal since the late-90s, early-00s.
A good website to keep track of films shot in Salt Lake and Utah is IMDb, which keeps an updated list of filming locations for thousands of movies.
http://www.imdb.com/search/title?locatio...%20Utah,%20USA&sort=release_date_us,desc
You can click on each movie and read about 'em.
There are still some bigger movies filmed in Salt Lake, though. It appears The Counselor, which is set for a 2013 release, is partly filmed in Salt Lake City and it stars Brad Pitt, Penélope Cruz, Javier Bardem and Cameron Diaz - though, I don't know the extent the city was filmed, or will be filmed.
But yeah, the last movie (non-documentary) that prominently featured the city, at least a film that was widely released in theaters, appears to be High School Musical 3: Senior Year, which came out in 2008 and then before that, Unaccompanied Minors in 2006. So, definitely fewer prominent movies are being filmed in the city. Hopefully that changes with the opening of this new movie studio.