I'll preface my response by stating I am not a historian. I am a mechanical engineer who designs heating systems in cool climates, and I have also retrofitted systems for pre-modern buildings. My knowledge of Roman-era heating systems is that they form the basis of one of the major types of heating systems still in use today: radiant heating.
The Romans, as early as the beginning of the Empire, recognized how climate affected building construction and orientation. The primary source for this is Vitruvius's De Architectura, a ten book treatise on architectural science and design as it stood in the First Century BCE. Chapter 1 of Book VI describes how buildings must differ between northern and southern climates: 0
For the method of building which is suited to Egypt would be very improper in Spain, and that in use in Pontus would be absurd at Rome: so in other parts of the world a style suitable to one climate, would be very unsuitable to another.
This is of course still true today. I design building systems in coastal Canada different than an engineer would design a system in Houston, TX, or a modern building in Rome. Air temperature, humidity, and solar radiation affect how buildings operate and how comfortable an occupant will be. That said, Vitruvius believed that an architect should primarily focus on three major themes: a building’s strength, a building’s functionality, and a building’s beauty. Human comfort was not yet a design concern. Plutarch even recalls a visit by Pompey to Lucullus where the former chastises the latter for having built a villa so delightful in summer yet so uninhabitable in winter.
The Roman experience in northern climates is, by this time, sufficiently advanced to understand that buildings in colder climates must be enclosed (as compared to the airy and open villas of Italy) and ought to have their major exposures facing the sun to increase passive heating. Vitruvius also recommends that buildings ought not be exposed to wind. Heat loss in a modern buildings due to cold air infiltration created by wind can account for as much heating energy as all heat loss through building materials combined, and that’s with poly building wraps and vapour barriers. Infiltration losses in a Roman stone or timber building would surely be the primary form of heat loss.
While Vitruvius understood the different building properties of various materials (described in Book 2), he doesn't describe any understanding yet of insulation. He also doesn't mention the insulation properties of earth that produced the semi-sunken earthen homes in northern areas of Britain and Scandinavia. While the style of buildings changes with geographic area, Vitruvius doesn’t recommend any building material changes between northern and southern climates.
Where the Romans really got lucky is a system of heating described by Vitruvius in Book V, the hypocaust, or the heated floor. The Romans heated the tile floor of their baths primarily for comfort. Tile floors were built upon small pillars of stone that created passages through which heated air from a furnace could pass. The hot air transferred heat via convection to the tile floor beneath the floor tiles. The tile floor was then warm to the touch, saving the Roman bathers from cold feet.
The Romans, for once considering the importance of human comfort, had stumbled upon the concept of radiant heating. Any time a surface is warmer than another, the warmer surface will transfer heat via radiation. A heated floor will transfer heat uniformly to all other surfaces (or occupants), providing a much more even temperature gradient than would be found with hot air convection from a furnace duct or an open fire in a hearth. Larger systems have been found that directed hot air through wall cavities, increasing the radiant surface and thus increasing radiant heat transfer.
As an aside, there is an INCREDIBLY interesting doctoral thesis by Taylor Oetelaar that used modern computational fluid dynamics (CFD) analysis to determine the architectural and engineering performance and indoor environmental conditions that would have been sustained by a Roman hypocaust system. He modelled the hypocaust beneath the Baths of Caracalla to determine how the system would have operated under different climactic conditions. He also reviewed whether or not the large window openings in the baths would have been glazed, and how the indoor conditions in the space would have changed if the windows were open or glazed.
About sixty years after Vitruvius, Seneca describes residential apartment in Rome being heated using the hypocaust. Once we reach Roman Britain in the Second Century CE, heated floors and walls are becoming much more common. The villa at Woodchester (built in the early Second Century CE), has evidence of heated floors and walls in nearly every occupiable space. Residential buildings within the military fortifications of Hadrian’s Wall have been found with hypocaust systems beneath residential spaces.
These systems were not perfect. The Emperor Julian described a visit to Paris where temperatures were low enough to freeze the river. He suffered through a few nights of cold, and finally asked his servants to activate the heated flues. He quickly gets a headache and became nauseous and sleepy, finally vomiting. The Emperor blamed excessive vapours in the room. He likely experienced carbon monoxide poisoning caused by poor sealing of the wall flues allowing smoke to enter the room. His Imperial successor, Jovian, may have actually succumbed to carbon monoxide poisoning when he died after reigning only eight months.
Radiant heating in modern times began with the classic steam radiators, but heated floors have become more popular in commercial and residential spaces. It is one of my preferred ways to heat a building, and I’ve designed systems using radiant heat ranging from snow plow garages to luxury homes. Our modern systems are water-based instead of hot air. The most common form is to embed plastic tubes within a concrete slab. A boiler provides heated water to pump through the tubes, heating the concrete, and creating that large radiant surface that once heated Roman feet.
Sources:
Bernan, Walter (1865). On the History and Art of Warming and Ventilating Rooms and Buildings
Bozsaky, David. (2010). The historical development of thermal insulation materials. Periodica Polytechnica Architecture. 41. 49. 10.3311/pp.ar.2010-2.02.
Oetelaar, T. (2013). An Innovative Investigation of the Thermal Environment inside the Reconstructed Caldaria of Two Ancient Roman Baths Using Computational Fluid Dynamics (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/24900 http://hdl.handle.net/11023/435
Siegenthaler, John (2011). Modern Hydronic Heating: For Residential and Light Commercial Buildings