First thing I thought of:
It moves in war,
Smoke is blown about from flame, engines, explosions, gunfire, and is spread by the wind. If the battle moves, the smoke will move with it.
Abhors the floor,
Smoke is carried by hot air, and therefore rises. Even in an indoor fire smoke is thinnest near the ground.
Takes many forms,
Depending on the source, smoke clouds, smoke plumes, puffs of smoke, walls of smoke, wisps of smoke....
And is a controversial norm.
We've all seen smoke. It's not unusual. If we do see it outside of a specific context such as a bonfire or BBQ, though, most people will get excited. This could also mean smoke from cigarettes, which is a common although decreasing habit that many people dislike.
I answered this before a few more specific clues were added, so I'll fill them out down here.
An encounter with it
Will leave one tearful
Smoke irritates the eyes, causing tears.
And as an omen
It makes many fearful
As mentioned in a previous clue, if people see smoke where they don't expect it's considered a bad sign. It usually means fire or destruction of some type.
Though as journeyers swear
It's useful where
One wants to prevent
Meat can be smoked to greatly increase the time before it spoils. In past times, this could be used for a journey as food that wouldn't rot before you reached your goal.