How do you deal with a house under negative pressure?

That’s two problems you ‘ve laid out- air pressure and dust. First, I would say that if you have an excessive dust problem that perhaps you need to have your ducts cleaned, or perhaps you need to replace air filters in your heating or ventilation system (read more here). There must be a source of the dust, we would happy to help further but first find out when the last time the ducts were cleaned and check the filters. Here is our page on balancing air pressure in a home -

House Air Tightness - Why Balancing Air Pressure is Essential

On air pressure, there is clearly a balance problem with an excess amount of air that is being fed to the master bedroom compared to the amount that is being removed. And I would assume the problem then regulates itself if the master bedroom door is open?

So a couple of questions – have you always had this problem or have you recently done anything different like changing the door, or installing a thick carpet?

The simplest short term solution we could propose would be  – keeping the door open a crack and allowing air to escape. Ultimately you need to provide a route fo the air to escape, one of your choosing or it will find one that is less favorable to you. A return air vent in the bedroom would be ideal, but short of that I would say a gap at the bottom of the door would probably help a lot, which would mean removing the door and trimming a half inch off the bottom. That way the air would feed the return vent just outside the bedroom door.

But the real solution for properly balanced ventilation in your home, would be to have an HVAC contractor come and look, and likely balance the system. Either way, when you supply air to a room with a closed door, you need to have a way (return vent or gab under the door) for air to escape or it will lead to exactly the problem you mention.