A technician will usually check the burner, pressure, expansion tank, circulator, and zone controls on a hot water system, or the water level, pressuretrol, and vents on steam. In an older house, expect questions about past work, since radiators and piping may be decades older than the boiler itself.
If the boiler is replaced, ask whether the new one can sit higher in the basement, on a raised pad or platform, especially if your lot is in a flood district. The Building Department can tell you the district from your assessors map and lot number, and it enforces the National Flood Insurance Program in town.
Oil boiler replacements need Fire Department permits; gas boilers go through the Building Department's online permit portal. Ask the contractor to file them and to tell you when inspectors will need access.