When you say furnace, I assume that it is forced hot air and not water (which would usually be called a boiler)? If so:
His comment might also refer to the heat exchanger or the sidewalls of the furnace. In older furnaces, oil is burned in one chamber of the furnace (where the burner is located), and the combustion gases are exhausted through a chimney or a pipe through the side of the house. Another chamber has an air intake (either from the house or from outside), a fan, and a circulation duct that distributes hot air throughout the house. The two chambers are separated from one another by a heat exchanger; intake air that passes over this exchanger is warmed before being distributed.
Newer furnaces use many of the same features, but may have both intake and exhaust combined to increase efficiency.
It’s important to know that the combustion gases from the burner contain carbon monoxide. Sometimes furnaces are located in basements and the moist air can accelerate the rusting of the heat exchanger. If the heat exchanger is breached, instead of being exhausted to outside, CO can be distributed throughout the house with the warm air which can be fatal. You should verify if this is what the service technician meant. If it is the sidewalls that are getting thin, then CO may escape into the basement or living space through the sides of the furnace, which is still not a good thing.
So, yes, it probably is a safety issue.