Will removing dead skin leave scars?

Depends. If you are just removing the very top layer of skin, then no. However, any injury into the dermis can leave a scar.
Usually, yes. But the depth of the dead tissue is the important factor. Superficial thickness burns, for example, do not leave scars (when properly cared for). Deeper thickness blistering burns can require skin grafts or heal like freddie kruger's face! full thickness loss require excision, and flap or graft reconstruction. Both leave scars, but plastic surgeons know how to minimize these!