Does a carpenter needs to be registered/certified in order to carry out fire door and self closing device fitting job? Or any carpenter can do this job to a legal requirement level for a HMO house

For an HMO you need to check it with the local council (housing department) as they are the enforcing authority for HMO registration however,

Article 18 of The Regulatory Reform (Fire Safety) Order 2005, any person assisting the Responsible Person must be a competent person which means having sufficient training and experience or knowledge and other qualities properly to assist in undertaking the preventive and protective measures and I believe that should apply to your carpenters/joiners.

Other persons who assist the RP have third party accreditation, fire extinguisher engineers, fire alarm engineers, fire door inspectors and fire risk assessors but I am not aware of anything for carpenters/joiners however it would be a good idea for one of the association to consider the possibility.