How are electric under blankets fail safe?
Since 2001, all electric blankets now have a safety mechanism that will kill power if it detects that the element is broken. Many are now operated using 12 or 24VDC as well, so there is less danger of shock if something goes awry; the control unit is also a DC power supply. 

If your dog wees on it and the element is not already damaged, nothing will happen, the heating element wire is insulated. But you can't really wash it, so don't let that dog on the bed!