Do tubes in a water tube boiler burn off after plugging them. Should the damaged section be cut off before firing the boiler. would the damaged tuber couse damaged to the other tubes is not cut off?

It can be left in there without any immediate issues. It will not effect the operation of the boiler. Depending on it's location in the boiler it may burnout eventually but if it is near the last pass it will more than likely stay intact as the heat in this section of the boiler is not hot enough to burnout a tube. The tube will start to change color and normally the metal will begin to delaminate and show visible signs of stress before it lets go. If it is loose and accessible you may want to remove it when your unit is down. This will prevent it from falling onto other tubes which should really hurt anything but if it is a long tube that you plugged it could damage eh refractory and/or baffles. Hard to say much more without any other info on the unit.