Imagine you are in the mountains and it is very cold. The walls are colder than the air, so air close to the wall will lose heat to the wall, and by losing heat it might lose its humidity (water condenses to the wall). Now I know this is bad for the paint on the wall. However, in some sense, it is heating up the wall, which will lessen the cooling effect of the air. Please tell me if my logic is correct. Also, if the walls are now wet, could they allow for more condensation? Or it is impossible because all the surface is covered in water? PS: House without heating nor insulation.
Certainly we could calculate the heat gain to the wall due to condensation (a glass of ice water will melt faster in humid climates due to the heat gain from condensation).
In practice, though, this will be insignificant compared to typical winter cooling loads. Also, you will never heat the wall above the dew point of the air (temperature at which water condenses out of the air), which is likely in winter to be quite low.
You will never be able to actually heat the room this way because to heat the room you need the temperature of the wall to be above the temperature of the air in the room (heat always moves from high temperature to low temperature). Once the wall is hotter than the air next to it, no water will ever condense from the air.
One consideration that is being overlooked in your description is mass. The mass of the wall and the mass of the cold mountain are substantially greater than the mass of air that is in contact with the wall.
Additionally, the air will have convection currents, which will move the airflow in such a way as to bring new moist air into contact with the wet wall. The wet wall also is subject to gravity, causing the water to flow downward, making available convection and conduction for the "incoming" air. There's really no limit to the condensation relative to "heating" the wall, if the outside temperature remains lower.
As long as there is a disparity of temperature between the outside and inside of the house, there will be airflow and depending on the relative humidity, condensation. If there is no influx of drier air to the house, the condensation may cease.