Trip Coil Monitor

What does it say in the user manual for the equipment?

Why not?

Are there no possibilities of the trip coil getting damaged during breaker closed condition as well as when the breaker is in open condition?

If you look carefully at the wiring of the trip relay, you will usually find that the monitor relay circuit is looped through all the trip relay contacts. It checks the supply and wiring integrity as well as the coil. This wiring is often extensive and vulnerable to being dug-up by metal thieves, accident, fork-lifts, aside from maintenance error, corrosion - all of which could disable a breaker and lead to a fault being cleared by off-site breakers causing outage.