The heat exchanger on a cast iron boiler is the essential core of the thing,(and practically all of it's mass.) It's a series of cast iron plates through-bolted together, with water passage and flue passage sides both full of features for creating turbulence and more surface area on both the water & fire-side of the exchanger.
If the controls are working it should at least fire up no matter how much gunk & crud there is on either side of the heat exchangers, but the burns might be very brief. If it's not firing at all, it's not a heat exchanger issue.
If it's firing for just a short time and turning off, it could be a number of things. If that's what's happening, observe it's behavior- is it tripping the high temp aquastat? Is the pump running? Is there yellow flames, maybe even lapping back a bit?
If it's sooted up on the fire side it's usually a symptom of incomplete combustion (which could be caused by obstructions in the flue path leading to improper air/fuel mixture). If it's heavily rusted up on the fire side it's likely that the returning water coming back from the boiler from the radiant floors is chronically too low, and the thing really is toast.
Corrosion & rust or soot-buildup on the fire side is usually accessible/visible if you disassemble the cabinet top, or can see past the burners with a mirror. It's usually possible to clean up the fire side with wire brushes, coat-hangers, whatever it takes to free it up and vacuum out the crud. (It's not an easy job if it's really jammed up.)
If it's all corrosion & gunk on the water side that is restricting flow you might be able to move some of it out with a phosphoric acid flush, and you may create other issues (freed up gunk jamming valves or ruining pumps, acid-destroyed seals or other system components, etc.) Large amounts of corrosion on the water side is usually an indication of oxygen getting into the system, which can be either from a constant input of fresh water due to water leaking out somewhere, or an improper type of PEX tubing in the radiant floors that allows oxygen to enter the system water via vapor diffusion through the wall of the tubing.