It might help if 'trash' was broken down into categories.  Obviously there is a difference between burning empty cereal boxes and plastic milk jugs.  It may well be that both are bad, but it might help if the people who are going to burn anyway could at least remove the most nasty combustibles.

In other words, for those will burn trash, what can they do to reduce the pollution?

Yeah, I think it would be helpful to categorize the types of trash.  Here in our part of Japan, we have to divide our trash into a number of categories, one   of which is "combustible," which is supposed to be composed mostly of food scraps and wood, small bits of paper (as most paper is recycled), that kind of thing.  We don't burn it ourselves, though.  I'd like to have a better understanding of what might be okay to burn and what definitely wouldn't (well--I know it's bad to burn any kind of plastic, but it's other things I'm not so sure about).  I'm also curious about beach bonfires, where people seem blissfully unaware of what they throw on the wood to keep the fire burning.

A characteristic of Homo S Sapiens is that he lives in symbiosis with fire. A characteristic of the Late Homo S Sapiens is that he doesn´t have any clue of real (open) fire any more, for the many fires he uses are hidden in black boxes. This is one aspect of civilization dementia. Sometimes they sit at smoldering logs, call that a camp fire, happily inhale the smoke, and happily forbid you to have a cigarette there. I´ve seen dudes carrying "paper" (i.e. sheet stained with industrial waste) out in to the wild, to use it for lighting their stinky campfire. Before they barbecue their würstels, the plastic wrap needs to be burned, of course. This is coprophagia (eating shit), a typical symptom of dementia.

So, dear Homo S Sapiens, please forget about lighting open fires, unless you intend to use your brain. Yet, even if you manage to you refrain from insulting our species at the open fire, it is not so easy to avoid coprophagia: Seen as a whole, i.e. as species-in-ecosystem, the major business of the Late Homo S Sapiens is eating/feeding shit - the shit produced by the ecosystem over millions of years and dumped as hydrocarbons.

It is illegal to burn trash in Indiana no matter where you live in the state.

Umbra's comments seemed to be discussing mostly the burning of trash.  What about yard waste?  Each year I pull out tons of non-native and invasive honeysuckle.  There is far too much to take to a landfill and it would take years to rot down if left in some kind of compost pile.  I know the burn is giving off some CO2 but I honestly don't know what else I can do.  Suggestions?

My husband and I are going to build a new off-grid home, and are planning on using a masonry stove, specifically a Tulikivi, if we can afford it. My understanding is that these stoves burn very cleanly, emitting very few toxins.  Wouldn't this work for cereal boxes, too?

Does anyone know out there know for sure?

Even burning yard waste emits plenty of carbon monoxide, CO2, particulates and even some carcinogens. (I used to work with the USDA Forest Service and remember hearing how harmful the smoke from forest fires can be, though not as bad as a back-yard burn barrel.)

Some garbage haulers take "yard waste" in a separate "trash" container or at the curb and compost it or deliver it to farmers or gardeners who will compost it.

Another option is to rent a chipper once a year and run your twigs and branches through. The resulting small pieces can be used as a mulch, or add them to your compost pile. (They compost a whole lot quicker than the large woody pieces you start with.)

If neither of these work for you, you might contact your local garbage hauler and ask when they are going to start collecting yard waste!