The answer is no. The printing device will let you know when it can no longer print because of toner.
Now to discuss this further: A little knowledge first on how it works. Simple language here. Toner is added to a developer unit which sits real close to the image unit, but the toner does not rub on the drum (imaging unit).
An image that you want to print is placed on the drum by usually a laser beam. The drum has a specific charge of electricity and now even different where the image is because of the laser beam. The image is such that it attracts the toner from the developer unit to electrically stick to the image. Just a note here if you run out of toner it wont print anyway. Why print an image with no toner? There wont be any image.
Next step is to get the image onto the paper. As the drum rotates in time it meets with the paper to be printed. The paper will pass under the drum so the image will go onto the paper. Under the paper but directly under the drum is another electrical charge unit that is much stronger than that of the toner stuck to the drum. It now is pulled to the paper, and now on the paper. The paper now makes its way to what is called a fuser or fixing assembly (same thing). The fuser now melts the toner and presses it into the fibers of the paper and then your print is done.
The process I explained is a little more involved and has a few methods it can produce the same effect but all the processes have toner a developer unit, a drum, transfer unit and fuser.
Now to answer your question. How can it matter if you are out of toner or not? The toner is not a lubricant for any component and no damage can occur with no toner. You wont get any print on your paper for one thing. The other thing is your printing device knows when it is low and out. The reason is not because it will harm your machine but because if your out of toner why would want to continue printing if blanks comes out? Waste of paper.