Energy efficiency in museums and buildings that house works of art or cultural heritage appears to be a difficult achievement if indoor air quality has to be kept at appropriate levels for artefacts’ long-term sustainability. There is a gap in our scientific literature on the relationship between indoor air quality and energy efficiency, meaning that there are no numerical data that examine both of them simultaneously, although this is a theme that is broadly discussed by museum managers, curators, and scientists. It is certain that the two parameters, indoor air quality (IAQ) and energy efficiency (EEF) are conflicting and difficult to reconcile. Furthermore, IAQ is not only the determination of temperature, relative humidity, and CO2, as is usually presented. Using green or renewable energy does not make a building “energy efficient”. Hence, in the manuscript we review the literature on IAQ of museums and exhibition buildings, in conjunction with the consideration of their EEF. Hopefully, reviewing the literature for this problem may lead to carefully designed monitoring experiments. The selection, application, and testing of appropriate technological measures can lead to a new balance between the two conflicting parameters. Not only must solutions be found, but these solutions are necessary in the mitigation battle against climate change.