I have a Krampouz crepe machine made of cast iron. Someone rested what looks like a hot mug of tea on top of it, and now it has a dark brown burn patch. Does this mean it will no longer work, or is there a way I can remove this?
This is not a burnt patch, cast iron doesn't burn. It is almost certainly a rust patch.
Since the surface comes into direct contact with food, you cannot use the rust removers sold in home improvement stores. You need to remove it with lye - you can try a mechanical stripping too, but that is unlikely to get all tiny specks of rust, unless you get serious about it and use a dremel attachment until you bite into the blank metal. See also my answer here about removing rust from cast iron.
Once you have stripped that part (or all of the seasoning, if you use lye) you will need to reseason. Since this is a crepe maker, you don't have to do much with multiple layers etc., just enough to seal the surface against rust until the next batch of crepes you make. Making more batches will give you an even better surface.
Above, I said "almost sure" that this is rust. Since this is only a photo, there is a tiny possibility that this is stuck-on dirt which looks like rust. If you think that this might be the case, you could try removing it with an abrasive cleaner, or with a metal sponge. But since this will be a lot of work and the chance is small that this is not rust, you can just treat it like rust from the beginning.