I have an account on a school system where they won't allow us to mount virtual images. I can run arbitrary programs as a user, though not as root. I've tried mounting it from the command line using diskutil and hdiutil (or somethings like that), and I have an application to convert it between .iso, .cdr, .dmg, and .img. I don't want to have to use Disk Utility to burn it to a CD each time to open it and don't want to need additional hardware (i.e. a CD-R or flash drive). I just want to get files off it. The particular image is Chrome for Mac.
Pacifist allows you to examine the contents of packages and archives, including .dmg files, and extract individual files. Unfortunately, it may require you to authenticate as an Administrator to do so.
I would advocate caution in proceeding. Be certain you're not circumventing the intentions of your school's system administrators even if your actions are seemingly allowed based on the restrictions imposed.