A wireless or hard wired doorbell?
If hard wired, it could be due to chaffed wireing that is short circuiting with the vibration of the heating system. There are (usually) three components. The Button, the chime and the transformer (which may be close to the boiler). Locate as much of the wireing that is viewable and look for wear and tear on the insulation.
I'd also inspect for dirt and corrosion at/inside the button(s) and chime.

If it's a wireless doorbell, maybe its noise rejection isn't very good. In this scenario, when it starts, the boiler produces electrical noise which the doorbell chime unit interprets as a signal from the pushbutton.

Try relocating the doorbell receiving unit (chime) to a different place in the house.

If that doesn't help, one solution would be to add noise suppression circuitry to the boiler. A cheaper approach: try a different brand of wireless doorbell system.

maybe the dorbell and the boiler share the same transformer, and there could be a mistake so when the thermostat calls for heat it ring the doorbell, too. i just had an idea for a prank!

Ok thanks. Yes it is hard wired.

This has been driving us nuts, and its time to have someone evaluate.

Its HALLOWEEN!!!