The growing sensitivity towards environmental sustainability, particularly in the light of climate change, requires a reflection on the role that historical buildings can play in heritage conservation. This research proposed an interdisciplinary approach combining climate and biological expertise to evaluate the biodeterioration risk associated with different IPCC outdoor climate scenarios. Conduction heat transfer functions and dose–response functions were used to model the indoor climate of a historical building and the related climate-induced risk of mould and pest proliferation. The approach was applied to a case study in the Mediterranean area, i.e., the State Archives of Palermo (Italy) housed in a 15th-century convent. In 2018, a survey conducted by ICPAL-MiC experts warned about past infestations and risks deriving from climate conditions. An environmental monitoring campaign conducted in 2021 allowed for the characterisation of the buffering effect in a historical building in response to the outdoor climate and the simulation of future indoor climate. Since indoor temperature and mixing ratio are expected to raise in future scenarios, it was found that there is an increased risk of insects’ proliferation, combined with a decreased risk of spore germination and mould growth. Such evidence-based evaluation allows for the design of tailored preventive conservation measures to enhance the durability of both the archival collections and the building.