Mears thermostat looks like model M70 or 7D - double line break.. in new rental unit making a head splitting high pitched whining noise every time heat is called for. It's an oldie. When I tap the metal sheet insisde the outer face plate it stops. Resumes again next time heat is called for.
Does she need to be replaced?

You might test to see how many amps are going through the thermostat to see if it is overloaded. Then if that is OK I would replace the thermostat.