Will Time Machine backup an iTunes library that's stored on a network share? I store my iTunes library on a Synology NAS and access it from an SMB share. The library is larger the SSD in my Macbook, so I cannot store it locally.  

Does Time Machine back a library that is on a network drive? 

I entered Time Machine today and tried to navigate to the iTunes library share but it isn't there. That leads me to believe that it isn't being backed up. 

If it isn't being backed up, is there a way to get Time Machine to back it up? If not, is there an alternative? If I remember correctly, Backblaze will not back up network drives. I'm not aware of another backup service for Macs, but I am open to suggestions. 

It's taken me countless hours to organize my library, fixing song/album info, download album art, etc, in addition to the cost of buying tons of music from various sources.
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.