It is not lime as in the fruit, it is lime as in the material (CaOH2). Lime the material is the binding agent in lime plasters. It is not likely to get washed away.
caoh2 is water soluble (slightly). the question was about anti-mold properties and it is commonly used for these properties in paint for example in cellars but it will lose these properties when too wet. so it's not used in bathrooms.
Sure but lime plaster can be used in bathrooms on the non shower walls as there's no running water but lots of moisture/humidity which leads to mold growth. As far as using it in the grout behind tiles, you may have a point.
I'm considering applying lime wash between the tiles regularly but not sure what the results will be.
Concrete is insoluble (it won't wash away, contrast with lime), but not water-proof.