Thanks! Geesh, I guess this means I have to start cleaning the house more...Even a non gaming computer will need a fan for the CPU and any graphics card worth adding will also have a fan.
There are some Fanless/Passive cooled cases on the market which basically use heat pipes from the CPU and sometimes even the GPU to the case itself which acts as a huge heatsink. The Streacom DB4 Fanless Chassis is a good example. There are others but Im not too familiar with them.
You can use a normal case which has filters to minimize the dust intake and then mod the filter even more buy slipping some fine pantyhose over them to provide additional filtration. Obviously you will have to maintain them by cleaning/replacing the filters.
You could also build a cabinet to put the PC Case in that has additional custom filters for dust that fits your needs.