For painting a thick fog on a canvas, oil paints are the optimal choice. By taking advantage of the slow drying time of oil paints, you can create a blended, murky image with softened edges and illusive forms, disappearing into the distance. Paintings of heavy fog will leave an overall impression of being light grey or light blue, so these paintings look best in rooms with cool or neutral colour schemes.
Paint the background with a flat wash of a very light grey or light grey-blue. Mix the paint for the background with turpentine to make the paint roll onto the canvas smoothly. Use a flat brush. Don't wait for the wash to dry before moving to the next step. Allow the paint from the background to mix with objects in the foreground, to add to the murky quality of the painting.
Paint your scene using light colours. Keep a large supply of very light grey and very light blue on the palette. Mix light grey to all the base colours that you would normally use on the painting. For example, if you are painting an image of trees, mix light grey with the green paint you'll use for the leaves and also to the brown paint for the trunk.
Objects in Foreground and Background
Paint objects that are closer to the foreground with darker colours and objects that are farther in the background much lighter. This means that you will add more light grey to objects that are farther away and less grey to objects that are close to the foreground. This is standard practice anyway, but exaggerated for painting a fog. Leave out details except on objects very close to the foreground -- the fog should obscure details on anything not very close to the front of the picture plane.
Soften the edges of the forms in the fog using a dry brush. Run the brush over the edges of the paint before it has dried, in order to create forms that have softer, less hard edges. Use a soft brush that won't leave brush marks in the paint. If you feather the edges of an object too much, wait until the paint has dried (this could take a few days), then repaint the edges and soften them again, but this time feather them less.