You usually don't need planning permission, but Building Regulations and smoke control area rules almost always apply to wood burner installations.
By law, Building Regulations Part J must be met for any wood burner installation, and in a smoke control area you may only burn an authorised fuel or use a DEFRA-exempt appliance under the Clean Air Act.
Listed building consent is a legal requirement for any work, including flue installation, that affects a listed building, inside or out.
Building Regulations approval is a legal requirement for installing a solid fuel burning appliance, even when planning permission is not needed.
The installation must be certified either by an installer registered with a competent person scheme such as HETAS, or by Building Control (the council or a registered building control approver).
Under the Building Act 1984, the council can require you to alter or remove non-compliant work, and you must be able to prove compliance when you sell.
In Scotland, check with the council's building standards team whether your installation needs a building warrant before work starts.