This is only a problem if your resources are organized to mimic a hierarchical system. Like a file system.
I prefer non-hierarchical resources. The "path" to a file would just be a property of the file. To copy-paste, there are two options.
- If you really just want another "path" reference, add another entry for the "path" property. The same exact file is "in" both "folders". 
- If you need to new version of the file, effectively forking changes thereafter, create a new resource (different URI) with a different "path" property. 
- To move, just change the "path" property. 
If you must insist on hierarchical, just mimic how a file system does copy-paste and move.
The copy is easy. A GET for the resource to copy.
To paste, a POST, because you are creating a new resource, a new URI.
If you need to do a move, you probably need to DELETE the old resource.
If you want, you can specify a location in the delete request, allowing the server to redirect users looking for the moved resource at its old location.