You missed the best and most environmentally sensitive yard clearing tool: goats. Google it. There's goat farmers, marketing themselves as green weed clearing, all around. They bring the goats to your place, the goats eat, they leave. Bing, bang, boom. I'm sure they'd love some honeysuckle.

Once upon a time, a long long time ago when I was young and innocent (and that was a long long time ago)I burned a pile of leaves and branches in my garden plot.  It was a dark and stormy night - the high school football field lighting across the street illuminated the smoke from my fire like London in the Blitz.  The local fire company stopped by to say "hi" and remind me that open fires are illegal in Montgomery County, Maryland, which was highly embarassing because I used to be on their governing Fire Board.  Anyway, my solution now is to make sure my yard waste (the stuff that cannot be composted within a decade or so, and avoiding poison ivy), is very dry.  Then, with other good wood or charcoal -I legally use it to cook my steaks!  Sometimes, it makes a very interesting taste combination.  and the Fire Department leafs me alone now.

I've yet to find a honeysuckle vine that won't succumb to mulching mode on my $99 Home Depot lawn mower.  Really, I barely even need a compost pile.  I have a small one for large branches that I eventually chop up and use strategically to stem soil erosion.  

Is burning gasoline in a lawnmower preferable to burning wood?

My head spins.  

If the stems are fine enough, an electric leaf blower with a vaccuum/mulcher attachment might reduce the stems to a manageable, compostable volume.  Otherwise, there are small wood chippers you can rent and use yourself.  Neither of these solves the "what to do with the resulting mulch" problem, but they do provide relatively low-fossil-fuel ways to make the volume smaller, which is a plus whether you're composting or paying someone to use fossil fuels to haul the stuff away.

A high combustion lawnmower engine, yes, burns much cleaner than a lazy yard burn pile.  

Who said a fire has to be "lazy?"  If it's maintained a little, it burns hot enough to incinerate most of the smoke.  If it's extinguished before it burns out, it sequesters some carbon.

I don't know what a "high combustion lawnmower engine" is, and I can't seem to find any info about them.  I maintain our 'lawn' with a a few grazing animals, and a scythe; and hope to never have to buy a lawnmower.  

I was just hoping for some balanced information about the emissions of different kinds of combustibles under various outdoor burning conditions.  Without that, I don't see how anyone can make a valid comparision between the various options.  "Much cleaner" isn't something I can quantify or verify.  

We've been wrassling with a burning dilemma of our own. Let's expand from trash fire to yard waste fire to...field burning. We have around 100 acres of fields that we transformed from cattle pasture to native warm-season grass prairies through a governement conservation program. We are required to burn one field a year (approximately 30 acres). That option is becoming less and less palatable, and when our government contract is up, we're left with a choice: do we continue burning, which is a more natural choice historically for managing prairies, or do we burn fuel and mow with a tractor?

Decisions, decisions...

You might want to talk with a restoration ecologist. Your decision regarding burning or mowing is going to depend on your local climate, what you want that pasture to look like, what you intend to do with it, and what sort of invasive plants are around it.

For example, burning is important for prairie ecosystems because it removes the litter and allows the ground to warm up early. Mowing will not accomplish this, though you might consider haying it and removing the material. This might be a compromise.

Why is burning becoming less palatable? It is a natural process that encourages the native grassland community. There is still a net gain in carbon storage in the soil... if that is what is bothering you... prairie plants put a large fraction of their carbon deep into the soil.

The reason towns and cities usually limit yard waste burning is because 1) they don't want to have to put out any fires that get out of control, especially in densely populated areas, and 2) the fine particles generated by low-temperature combustion are pretty dangerous - aggravating asthma, increasing long-term cancer risk, etc.  A "well-tended" open fire still will produce lots of particulate matter; the average home wood-stove produces a lot even though it gets much hotter and more evenly hot than a pile of burning brush ever could.

In terms of carbon, whether you deal with yard waste by burning or composting, most of the carbon will eventually end up back in the atmosphere, where it came from in the first place.  Any charcoal produced in parts of the fire that didn't get enough oxygen is almost pure carbon, in a form that will take a very long time to return to the atmosphere.  However, the vast majority of C in burning yard waste is released to the atmosphere immediately.  If you compost, you are keeping the carbon sequestered for longer (the end product of your compost pile is about 50% C dry mass, and will continue to decompose many years after you put it in the garden).  Ironically, the best way to sequester the C in yard waste would be to put it in a landfill - without sufficient oxygen, decomposition will take a very long time.  However, most localities wisely prohibit disposing of yard waste in costly and rapidly-filling landfills.

The bottom line is that a properly maintained compost pile is by far the lowest-impact way to deal with your yard waste.  If built properly, it will kill weed seeds nearly as effectively as a fire would, and it saves you from having to breathe in all that soot.  As for honeysuckle, be very aggressive with it starting early in the season, and you won't have such a huge unsightly pile (full of seeds!) to deal with in the fall.