Heat oil in a large Dutch oven over medium heat. Add mushrooms to pan; cook 5 minutes, stirring frequently. Add onion; cook 6 minutes. Add celeriac, carrot, and parsnip; cook 4 minutes or until onion is tender, stirring occasionally. Stir in tomato paste. Add 7 cups water and beer; stir well. Reduce heat, and simmer 5 minutes. Stir in cabbage, potato, garlic, and beets; bring to a boil. Reduce heat, and simmer 20 minutes or until vegetables are tender, stirring occasionally. Remove from heat.
Place half of beet mixture in a blender. Remove center piece of blender lid (to allow steam to escape); secure blender lid on blender. Place a clean towel over opening in blender lid (to avoid splatters). Blend until smooth. Pour into a large bowl. Repeat procedure with remaining beet mixture. Stir in vinegar, sugar, salt, and pepper. Ladle 1 1/2 cups soup into each of 8 bowls; top each serving with 1 tablespoon sour cream and 3/4 teaspoon dill.
The simplest way to peel celeriac is to remove the rough, knobby skin with a sharp chef's knife.