Textures disappear from files when if you move or delete any texture from the relative and absolute paths of the associated media files location. This can happen when you give the FBX file to someone using a different computer who does not have that texture in that your directory on the your computer drive.
Use the Embed Media option to make the FBX plug-in includes (embeds) the associated media (textures, for example) within the FBX file. The plug-in then extracts the media to an .fbm folder at import. The plug-in copies the embedded media to a <fileName>.fbm folder in the same location as the FBX file and is named after the FBX file.
If you disable the Embed Media option, the FBX plug-in stores the relative and absolute paths of the associated media files at export time. This causes problems if, for some reason, the path is no longer accessible. Make sure that the associated media is accessible to ensure the proper import of these media files.
If you export a file to FBX and do not activate the Embed Media option, no problems occur if you open the file on the same computer and all the images used in the scene are in the same locations that they were when the file was exported.
It is a good practice to embed media when you are going to give the FBX file to another person. It is also a good practice to embed when you use FBX for storing assets, as you may eventually delete the reference images or move all the files to another drive or computer with different paths.
Embed Media is active, but I still do not see my textures
In previous versions of the Maya FBX Plug-in, when you saved an FBX file with the Embed Media option activated, the embedded files were extracted to a folder in the same location as the FBX file. If you did not have write permission for the folder, then the media files were not extracted. You can experience this lack of write permission when you work over a network.
With the current version of the Maya FBX Plug-in, if you do not have write permission for the folder creation, the media files are sent to a temp folder instead.