I ran out of oil in my home, and a plumber friend said that I could add diesel from the gas station while I wait for my oil delivery. Is he right?
Yes, diesel will work fine in an oil heater.
But don't try to put heating oil in a diesel vehicle.
It's illegal and will get you slapped with a $10k fine if you get caught.
Heating oil has a horrible cetane rating (in addition to lack of detergents and poor lubrication properties, and possibly higher sulfur content) and will make your engine run horribly and may damage it.