Before we get to the symbolic meaning, we can and should start with the literal meaning. This is a dirty place—a dirty world. The Party may be in charge, but they aren't doing a good job of things, and everyone should be able to see that from the dust on all kinds of things.
After that, dust has several meanings. It represents the inability of the party to keep foreign particles (ideas, desires) out. It represents the way the past stays around: everywhere, but in fragments. It represents the fatigue of the common man, ground down by the oppressive regime. It represents how the world marks us in faint but ongoing ways.
Dust is dirty and unwanted. It represents Party oppression for this reason. The Party, as O'Brien writes, tries to oppress its people by keeping a poor standard of life. This is why there is so much dust in London. Winston has to blow dust off his speakwrite before he uses it, despite the fact that he uses it everyday. Dust is often caused by war machines, showing directly how the Party causes dust to be in London. Dust shows stress on Mrs Parsons face as she is too occupied with her troubles at home to concern herself with her appearance. However, in Part II, references to dust are significantly different. They illusrate how Julia and Winston are temporarily overcoming Party oppression are living for once (even though this means bringing their deaths nearer and nearer). For example, Julia and Winston kiss through dust and Julia traces Winston a map to a meeting place in dust.