Short answer: No.
A laser printer will have a mixture of toner and developer in its developer housing. The physical interaction between the particles of toner and developer are what gives the toner an electrical charge. The polarity of this charge will depend on the development system being used in your specific printer.
The printer will either have a Toner Concentration (TC) sensor, or use an algorithm to determine how much toner is in the housing, and how much to dispense.
In many home and small office printers, when you replace the toner you also replace the developer as well, starting you with a known TC. If you are actually adding toner, only, then the printer will dispense toner to reach the working TC. I suppose it might take longer to tone up if you really ran it down.
In no case would you run into a permanent damage situation.