The risk from asbestos relates to fibres entering your lungs. You could live in a house, all of your life, with asbestos in the wall cavities and ceilings without any problems, as long as nobody drilled into (or otherwise abraded) the asbestos panels (which would release fibres into the air).
I can think of no reason why a boiler would have asbestos inside it and, even if it did, the fibres couldn't become airborne. Thus, it seems reasonable to assume that the fitter is either referring to an asbestos jacket around the boiler or the presence of asbestos in the area immediately surrounding it. (i.e. it's unlikely that the boiler itself actually 'contains' asbestos but there could be asbestos in the surrounding area).
You need to seek further information about the risks which the fitter believes he's identified. Then look under 'Asbestos Removal', in Yellow Pages to seek a (free) quote for the removal of the jacket (or whatever else is concerning the fitter).
Chris