Is it possible to back up short messages from an iPhone on Linux, without iTunes?
Yes, you can use the libimobiledevice project to backup your iPhone.
The project website is: http://www.libimobiledevice.org
However, most Linux distributions have it available in their package managers for easy installation.
After you have libimobiledevice installed, run idevicebackup2 to take a backup of the short message and other data from the phone:
idevicebackup2 backup myfolder
where myfolder is a path to a folder, where you want to store the backup.
If you neeed to enter passwords, you'll want to run it as:
idevicebackup2 -i backup myfolder
-i means to ask for passwords interactively.
To then browse the backup in a traditional file tree format run:
idevicebackup2 unback myfolder
idevicebackup2 -i unback myfolder
A thorough guide can be found at this link.