The df -h command should be used to identify the issue of not being able to save large files to a directory on a Linux server. The df -h command displays disk space usage in human-readable format for all mounted file systems on the server. It shows the total size, used space, available space, percentage of use, and mount point of each file system. By using this command, a technician can check if there is enough free space on the file system where the directory is located or if it has reached its capacity limit.