If the substance can be cut it will be cut by this.
What you are describing is commonly referred to as a Monomolecular blade. The blade is so thin that it is either one molecule or less than three molecules wide. Most attempts to make a blade like this fail since any blade that thin is inherently very fragile, but you side step that problem entirely. Essentially if the walls are vertical, the tops of the wall are all monomolecular blades at the very top.
Sure but anything?
Theoretically, monomolecular swords can cut through molecular bonds if they are thinner than the bonds. However, there are cases where the wall will be thicker than individual bonds. In this case we use traditional methods of determining how effective it is at cutting. To determine if a material can be cut by a blade you need to know the strength of the material and the pressure applied by the blade to the material, as well as how much the blade bends. Since the walls are immovable there is no give to the wall so we can calculate the pressure in an ideal scenario without taking into account bending. Next we get the pressure. If an object rests on the wall, the force applied to the object in one G gravity is 9.8*Mass Newtons. Assuming that the atoms in the wall can get up to 1.5 nanometer wide for large rock molecules, making the maximum width about 4.5 nanometers. So the area the pressure is applied over is the length of the item across the wall time 4.5 nanometers. This will make the pressure incredibly high and let the wall cut through basically anything.
Diamond is very tough. It has a density of 3.51 g/cm^3. Placing a cube 1 cm on a side on the wall would give it a weight of 3.51 g and a length of 1.41 cm along the edge of the wall if we put opposite edges on the wall. This means the force on the cube is 9.81 * 0.00351 newtons or 0.0344 newtons and the area of the force is 4.5 *01^-9 * 0.0141 meters or 6.345 * 10^-11 meters squared. Since pressure is force/area the pressure is 3.44 * 10^-2/6.345 * 10^-11 pascals, or 5.42 * 10^8. This pressure is comparable to a high end water cutter without the use of any energy aside from the gravity applied to the material. Water cutters can cut through diamonds. Therefore, the wall will cut through the diamond.
Can it cut through large objects?
In most cases yes, but in some cases no. The wall will support some of the weight of the item with the edge of the blade through the pressure on the blade. If the item then is put on the wall in a way that it can use friction on the wall to support it it can reduce the weight centered on the blade edge, reducing the pressure to a point where the blade won't cut anymore, or at least without more energy put into it. he larger the object, the higher the chance this will occur before it cuts all the way through. If you cut through butter for example, cutting straight through is easy, but if you turn the knife as you cut the butter becomes harder to cut through since you have to push the butter aside with the knife. With harder materials this will become more noticeable.
The 4.5 nanometers is not a hard number, The wall would likely be thinner that that in most places, increasing the pressure up to 40 times if the wall is a single hydrogen atom wide. While on average the average might be only 2 nanometers, this doubles the pressure. Any item more vertical than it is horizontal will also have increased pressure. Any item that puts more than one G of force on the wall will have more pressure, if an item falls unevenly on the wall since the wall does not bend there will be more pressure on the one spot that hits the wall. Even after hitting the wall the item might not slow down as the wall can only impart a very small amount of force as the object passes due to the low surface area, leading to the rest of the object to hit he wall at a higher speed than a wide cutting implement.
Simply touching an object o the wall won't cut it (literally and figuratively). Some force must be applied for the cut to occur. If you put steel on the wall with less than one G of force under certain conditions it will not cut since the pressure is not sufficient. However, since simply letting an object rest on the wall will cut most materials this is not a huge problem.
Since you have already stated the existence of immovable objects, there is something this wall can't cut. Armor made out of immovable objects. The wall is immovable, but the wall is actually moving all the time. The planet moves under it, and the solar system moves around the galaxy. The wall is immovable in relation to something. So if you make a small immovable wall that is immovable in relation to your arm you have a perfect armor.