For an oil boiler, a technician typically checks the burner nozzle, filter, electrodes, and combustion, then the controls, circulators, and expansion tank. For gas, they'll look at ignition, gas pressure, and venting. In an older house, expect them to ask about the chimney, since many oil and older gas boilers still vent through it.
If the boiler has to be replaced, oil equipment and tank work go through the Fire Department, while gas piping is permitted and inspected through the Building Department's plumbing and gas inspector. Moving from oil to gas or to a heat pump changes which permits apply, so ask the contractor to list them.
Access can be tight in antique homes with low basements or fieldstone foundations, so old boiler removal may take more time and hands than you'd expect.