Andi,
If your monitor is a CRT type (big and thick) the life expectancy is close to 5 years when left on 24X7. I have been in the computer/tech business for 25+ years and have never had to replace one because it was left on. I have left every PC and monitor I have ever owned on all the time because I don't like to wait (even 10 sec.) for it to "wake up."
The real danger is electrical surges so you should make sure you have at least a surge protector, but better to have a battery backup.