What risks may arise from putting a capacitor on the supply line to the door-bell?
Don't do anything like this. Mains voltage is not a toy and it's not for tinkering.  Everything that touches mains voltage should be UL/TUV/etc.* listed as mains equipment and installed according to its labeling and instructions. Bare components from the electronics supply are not equipment: they lack labeling and instructions for use in mains wiring, and cannot be so used. 

It's very likely the device failed not because of over-use, but because of defectiveness.  Running a buzzer til it's burnt out won't damage wires unless the buzzer melts down and shorts out.  You may have dodged a bullet as far as your house not burning down.  

I would go to a low voltage DC doorbell, because low voltage DC is treated much more lightly by the electrical codes.  

Either get a (very, very small) solar panel, battery and charge controller, or kitbash a consumer product that already combines those, such as one of those solar path lights.  Or get a low-voltage "wall-wart" transformer.  Have that power a low-voltage doorbell via the button obviously.  Ordinary use won't flatten the battery, but extraordinary use would.  



* CE is not a testing lab.