You could load a request logger extension in your browser and visit all the pages of the live site, then check the list for any dev.example.com occurrences (or whatever the dev domain is.) A Firefox extension for example HTTP Request Logger, but this might be better for double-checking checking specific page changes as it would be fairly time-consuming… But it would pick up for example relative paths in stylesheets that point to resources that do not exist on the live site.
Why not just search the live database and all it’s files for any reference of the dev domain? There are database search plugins available to provide a UI for this you would probably just be able to write some simple directions for the other staff to use it.
For the files, it is less likely to be necessary as you would probably not hardcode dev domain URLS in your code but it may be worth a checking (using grep via SSH for example) if it is a major dev update or taking over from another developer.
Another idea… make a copy of the live site locally instead of the dev site and somehow block access to the dev domain temporarily from your local access, ie. client-side not server-side. Then browse your local copy, you would soon see any resource problems in your console window.