Will Time Machine backup an iTunes library that's stored on a network share?
No, Time Machine does not backup network drives.

You would need to setup a system to copy/sync the network drive contents to your own computer in order to have it backed up by Time Machine. This means that you would need space on your own computer to actually store the whole contents of the network drive. You can use the program "SyncTwoFolders" to achieve this.

The most common method though of backinging up Synology NAS drives is simply to use the backup solutions available for the Synology NAS itself. Depending on the model, you'll often have Synology HyperBackup available, and you'll be able to backup to local externally attached hard drives, to S3 in the cloud, or a number of other backup destinations.