Contact your IT people. They may have done something to the network that you are unaware of.
Check all of your IIS folders (inetpub/ and below) for any .htaccess files. I would do this through Windows Explorer (not Dreamweaver) making sure to check for hidden files.
If there is nothing there it is probably trying to include content from a password protected directory somewhere. Check your site code (including any Includes, CSS, JS etc) for any external links. If necessary comment the links out a and try the site again.
Look in your site definition and tell us what is the path to the default images folder for the site? I wonder if that is specified pointing into a password-protected folder.
Resolved the issue by setting my view to "Design" instead of "Live."
Our client's site requires users to be logged in, and the files I was working on do call out to the client's servers/databases. These calls (which went out every time I made a change like inserting an image) were triggering the popup. Setting the view to "Design" instead of "Live" prevents the calls from going out. Hope that explanation makes sense. I only work on content for a specific portion of the site (which is massive) so most of the code was not written by me, nor do I understand it very well.
I used TwoSuits suggestion to comment out external links to determine which one was causing the pop-up, and then commented out various .get functions inside that file to figure out the exact URL. While this in itself didn't fix the problem, it led to a coworker remembering the trick to set the view to Design (which I had done when I first started work ~3 months ago, but a recent Dreamweaver update erased all of my settings).
Thank you all for the replies!