This sort of thing doesn't just happen out of the blue. In this case, what has happened is that a user has opened an .xcodeproj and is unaware of the difference between the various relative or absolute Path options that a Group can be a part of. A Group in XCode is always shown with exactly the same manilla color folder-icon inside your XCode project, no matter what modes or properties it has defined inside of it.
By default the simplest case is that you create a new folder Group object in XCode and it's purely a cosmetic organizational tool that has no disk location information stored in it.
This is not the ONLY thing that these groups do, and not the only "mode" that these groups can be used in. These groups can also be used to point at some folder and say "things that are in this virtual folder are really somewhere else, either underneath this project's main folder in a subdirectory, or even up somewhere else on your hard-drive, either stored in relative path, or absolute path format". When used like this, these things remind me of a Windows "Shortcut" object on the desktop, or a Mac "Alias" object in the finder.
Dragging a file from one group to another does not move it to a different folder on the disk. It simply moves a reference to a file with a certain name, to another group, which might mean that after you drag a file, you haven't really moved it, or copied, or relocated it in any way, you've just moved an alias from a place where it could resolve properly to a real file, to a place where it can't. Thus XCode helpfully turns it red for you, without any helpful error message about what happened.
How do you fix it? In this case, go to the Identity inspector pane in the Utilities menu, and either decide to clear out the bogus value in the place where I have shown in the picture in the original question where I had "Classes". Clearing out a value that is invalid is not exactly easy to do because XCode requires that you basically find the root folder of your project and select that, and that will 'clear" the relative or absolute path property on your folder-group.
Alternatively, you can leave the folder alone, and just don't drag files from group A to group B without first checking what relative or absolute path they reference.
What still seems horrible to me is that XCode tutorials tell you to "use XCode to manage your project's contents, don't just drag files around in the finder inside an XCode project directory", and that's good advice, but it leads me to assume that XCode provides full and intuitive physical (and virtual) group-folder organization tools. It does not. As an example, imagine you inherit a project that has .m and .h files scattered through four physical folders underneath the main XCode project folder and you want to move those files around. You have to do a combination of tricky things inside XCode, and either in Terminal or in the Finder, in order to reorganize your folder. With the complications involved in moving items around in your version control tool of choice added upon the top of that, and XCode's very limited support for only Git and Subversion, you have a really tricky mess.