This problem is something I have come across before, but I'm not sure of the details. What I remember is...
Something about the pump drawing too much current at startup, and welding the pump contacts in the boiler. The pump manufacturer supplied a fix for it.
OP - what pump make and model is it? Try this and see if it provides a temporary fix, for one cycle of the boiler, at a time...
Turn the boiler off, give the boiler case a bit of a thump, with the flat of your hand. That may/should release the welded contacts. Turn the boiler back on, and then check if the pump also runs.
[EDIT] I've just remembered a little more - it's the back EMF from the pump at switch of, which welds the contacts up.
Do a search on your pump make/model plus back EMF.