They are a fire hazard, as well as possibly causing illness through leaving droppings, mites and parasites. Removal of a bird nest is advised as soon as possible and needs to be performed by a competent professional, with the correct knowledge and equipment.
Under no circumstances should anyone attempt to clear a bird nest through lighting a fire, as this could lead to serious complications: filling your home with smoke, or worst case scenario, causing a chimney fire which could then spread. A professional removal of a bird nest uses specialist equipment to ensure complete safety during and after the process is complete.
If the bird nest is already occupied however, it is an offence to remove it. This is due to the Wildlife and Countryside Act 1981 and the Protection of Birds Act 1954. These are the primary legislation protecting animals. Plants and certain habitats in the UK. The law states that all birds, their eggs and their nests are protected, any damage or removal of these can lead to penalties such as fines, or more severely, prison time of up to 6 months.
Removal of a bird nest cannot occur while it is in use or being built, so you will have to wait until the fledglings have flown away.
After the nest has been removed, I will do a full power sweep to ensure your chimney is fully clean and ready for use.