I use BackInTime (which is basically a front end for rsync) for backups, and I run one every night at 1 AM. This is on Linux Mint Cinnamon. If the computer is locked/the monitors have gone to sleep (computer isn't suspended), when the backup begins the monitors turn on, and will then stay on all night. I don't want to waste the power or wear out my backlights.
How can I stop it from turning the monitors on, or how can I get it to turn them back off?
Sounds like it's probably trying to inhibit sleep but uses the wrong way to do it that's meant for video playback and games and it just wakes everything up. Look for a way to disable that for this specific app.
I guess you can also
while true; do xset dpms force off: sleep 5; done
But that's just bleh for a solution.
If it can't be turned off, I'd just shove it into a container to isolate it and prevent it from doing that.