I will try to answer these issue intellectually. I trust you can follow.
First, you are right. Poor people do not have a lot of options. To keep warm, to eat, or survive. But I doubt this is fully true of the OP. The OP is burning unseasoned wood in an area that has regulations about burning. So the answer to your question of "What gives the neighbor the right to tell him he can't", is "the local law and regulations may tell him". One may not agree with laws, or regulations. Some are fully unjust and silly (e.g. Indentured service and slavery was once legal). But a legal law based system is how most societies work, right or wrong. And, equally, the OP has the right to fight the complaint. Which is also his legal right.
Second, simply because something is done commonly or historically in some area does not mean it is smart or good. I have the opposite problem of the OP. Most everyone around me burns unseasoned wood.
My chimney during burning:
View attachment 237705
Everyone else burning where I live:
View attachment 237706
Yes, many do this because they are poor and can not buy wood "years ahead". But they are also rather stupid. Not really understanding the concept of seasoned wood because buying unsplit logs in the fall and burning them that winter is "that is the way they have always done it". Simply buying and splitting in the spring would be a leap forward. But no. They are uneducated and stubborn, and many seem to want to stay that way. And that refusal to change is sad mostly because it harms them more than anyone else (I, fortunately, live above the inversion layer, and most days I can see this pollution, but don't have to smell it -- but if I go down there, the aroma is breathing in pure creosote). Where I live, surprise to no one, has a high incident of lung cancer, et al. But that affects everyone -- raises health care costs. So others ignorance and behavior ends up affecting me. The idea "I can do what I want on my property" has limits. Some things one does affects others both personally and financially. You don't live on an asteroid alone. Your actions do have consequences, even if you want to think they don't.
Side note: others in this smoke hazed town in the photo above would be willing to change, if they could because most of them have children and are of a younger generation (-- the local old farts seem too stuck in their ways and reactionary to change even if it helps them). But even those that want to change and help themselves need both education and support doing so (i.e. help buying wood a few years ahead). The local government has "issues" about that because those currently in power are incompetent. So it is complicated.