I live in a hundred year old home. I trusted a "professional" to take a wall down that was in between two trunk rooms. I asked to tarp the areas affected specifically spelling out where to put them. I came home to find my ceiling fans all on plus it looked like it snowed in my kitchen. Turns out the fans blew the dust up into my vents and then travelled all over the house. I bought a DUSTLESS Vac with a HEPA filter when the situation became desperate. That has helped but there seems to be a fine brown layer that is very hard to get up. Is this something that will go away with time as I continue to clean?
All that you can do is wipe everything down until it is gone. Problem is if this was a "handy man", you might as well get some help to clean up the mess, or have them come back and help, so you are not doing it yourself.
Even if you tarp, make sure the furnace is off or has a filter over the return, with it closed in that space, dust still seems to find its way everywhere. You are really talking duct cleaning, cleaning the squirrel cage of the furnace, which is the blower, the a-coil if it got dust all over it, due to no matter how good the filter is on the furnace, dust will get through it.