It sounds like you are basically asking about how water boiling kettles often get coated with calcium carbonate when they are used with mineral-rich water. It turns out that the calcium carbonate is actually less soluble in water at higher temperatures, and so it tends to precipitate at higher temperatures.
The minerals in the water will always be left behind. So the strict answer to your question is that the same amount of minerals will be left behind regardless of the temperature of the water or how quickly it evaporates. However, I think that the way that the calcium carbonate is deposited might depend on the temperature or the roughness of the surface.
Crystals tend to form on rough surfaces, so, if you heated the water, causing the calcium carbonate to evaporate, then it would likely to form crystals on the rough surfaces. If you let the solution evaporate at low temperature, then it would leave behind the calcium carbonate on all surfaces.
This question has a good description of the situation with the kettle: http://chemistry.stackexchange.com/questions/4094/white-powder-observed-after-boiling-water-in-electric-kettle-for-many-weeks