Undoubtedly, heritage buildings serve as essential embodiments of the cultural richness and diversity of the world’s states, and their conservation is of the utmost importance. Specifically, the protection of the structural integrity of these buildings is highly relevant not only because of the buildings themselves but also because they often contain precious artworks, such as sculptures, paintings, and frescoes. When a disaster causes damage to heritage buildings, these artworks will likely be damaged, resulting in the loss of historical and artistic materials and an intangible loss of memory and identity for people. To preserve heritage buildings, state-of-the-art recommendations inspired by the Venice Charter of 1964 suggest real-time monitoring of the progressive damage of existing structures, avoiding massive interventions, and providing immediate action in the case of a disaster. The most up-to-date digital information and analysis technologies, such as digital twins, can be employed to fulfil this approach. The implementation of the digital twin paradigm can be crucial in developing a preventive approach for built cultural heritage conservation, considering its key features of continuous data exchange with the physical system and predictive analysis. This paper presents a comprehensive overview of the digital twin concept in the architecture, engineering, construction, and operation (AECO) domain. It also critically discusses some applications within the context of preserving the structural integrity of architectural heritage, with a particular emphasis on masonry structures. Finally, a prototype of the digital twin paradigm for the preservation of heritage buildings’ structural integrity is proposed.