Birds Nesting in Chimneys can cause a lot of trouble.
As far as a chimney may be concearned; it can be condemned from excessive damage.
For the occupants of the house; they may encounter health problems ranging from Carbon monoxide poisoning if they are using the appliance with a nest restricting the flueway, Avian / Red Mite infestations and in rare cases the risking of catching fungal diseases, ticks or dangerous parasites know to travel on birds.
Generally, in the UK, the culprit will be the well renowned and very busy Jackdaw; often filling chimneys from bottom to top with anything they can find and everything that you dont want to find.