This paper investigates how the heating sector can best support the green transition into climate-neutral societies. In order to do so, the heating sector must be considered and analysed as an integral part of the entire energy system as well as be coordinated with other greenhouse gas emitting sectors. Consequently, in this study with Denmark as a case, we establish the context of a full transition of all sectors into a climate-neutral society. Using such context, we investigate the role of the heating sector with a focus on excess heat potentials, energy savings vs. supply, district heating vs. individual heating as well as the ability to recycle low-temperature excess heat and to provide flexibility and support the integration of variable renewable energy sources also in other subsectors of the energy system. The results of the analyses show (1) a feasible reduction level of around 36% in end-use heat demand in buildings, with higher potential in older buildings and lower in new buildings, (2) expanding district heating to areas with a density of 15-10 kWh/m2 is feasible, resulting in a district heating share of 63-70% of the heat demand compared to the current level of 51%, (3) a large unexploited potential to use heat sources such as industrial excess heat, geothermal heat, data centres and power-to-X, reducing the need for biomass in the heating sector, and (4) increased use of variable renewable energy and combined heat and power production increasing gas export and thus replacing biomass consumption outside the country.