Is it possible to run [ tmsh save /sys config file ] where [file] is a remote directory?

The easy way would be to go with a bash script and run it on cron if you want automated. But to send the file to some remote server passwordless [article], you would need SSH Key exchange done to avoid password prompt.

!/bin/bash

tmsh save /sys config file backup.scf.$(date +%Y%m%d)

scp /path/to/backup.scf.$(date +%Y%m%d)user@server/f5/backup/backup.scf.$(date +%Y%m%d)