Putting on the heating can work to prevent damp in certain circumstances. However, just because your house is warm, does not mean that you will not get damp. Adequate ventilation is still really important if you want to avoid damp. Interestingly, there is one specific type of heating that will actively fight damp, which is infrared.
Many people have turned to using infrared panels to heat, and these panels work through the use of safe radiation. This means that they are able to heat the fabric of the building, and the people inside it, directly. The heat will be stored in thermal mass, even when you turn off the heaters.
Infrared heating can help to dry out the walls if they have previously been affected by damp, providing that there are no other underlying issues. They can work to ensure that the walls do not get cold enough for water to settle on them.