That is not true unless the phone specifically has an app that remains in communication with the tower when the phone only appears to be off. Apple does stuff like that with “Find my phone”. But even that can be turned off in the settings. If the phone is not communicating with the tower then it cannot be tracked.
One thing to check is if your charge percentage decreases even when you think the phone was off. So you shut the phone off with 83% and you turn it back on a week later then it should still be 83% or very close. As any battery may not be perfect at holding a charge. If it is at 75% or less then I would be very suspicious.
The phone is always trying to connect with a tower.