A technician will generally check the pressure, the expansion tank charge, the circulator, zone valves, and relief valve on a hot water system, or the water level, pressure control, and low-water cutoff on a steam boiler. On the burner side, that means cleaning and a combustion test, with a CO check at the boiler and vent.
If replacement comes up, the fuel decides the permits: gas piping and gas boiler installs go through the Building Department's plumbing and gas inspector, while oil burner and tank work goes to the Fire Department. Building permit applications can be filed online, and the Building Department doesn't inspect on Fridays.
Older basements can make removal the hard part. A heavy cast-iron boiler may have to come apart in sections, and a new high-efficiency unit may need a different vent path or a chimney liner, so ask the contractor how they'll handle both.