Let's say the octave script you are working on is located in ~/Documents/script.m . On save, you want it to be automagically executed by octave, including persistent figures.

Let's create a watchdog script ~/Documents/watchdog.sh:

while true
  ATIME=`ls -al ~/Documents/script.m`
  if [[ "$ATIME" != "$LTIME" ]]
    echo "Running octave script"
    echo "script" > ~/Documents/OCTAVEPIPE
  sleep 1

Make it executable: chmod u+x ~/Documents/watchdog.sh

Don't launch it yet. We are missing the OCTAVEPIPE pipe. Create it with mkpipe ~/Documents/OCTAVEPIPE

Launch the watchdog in a shell: ~/Documents/watchdog.sh or ./watchdog.sh if you are in the same directory.

In another shell, cd to /Documents/OCTAVEPIPE) | octave --interactive

Now edit and save your octave script, it shoud be executed after less than 1 second.