Out of boredom, I decided to shoot a civilian in the head, and was surprised that he actually died instead of ignoring it. Is there any reason for me to either kill them all or to refrain from doing so?
I ran through the game once on Normal, and was reasonably nice to everyone I met and avoided conflict.
I am now running through on 1999 Mode, and MURDERING ALL THE THINGS.
There's very little difference - even if Elizabeth is present and I'm skyhooking some poor woman's face in, she doesn't seem to mind. Despite her empathy towards the downtrodden in certain locations, I can murder them up with no difference in her demeanor. Collateral damage from a firebomb into a mixed group of enemies and innocents also doesn't seem to faze her. She just sort of nonchalantly leans against a wall as their faces melt off.
There are a couple of QTE encounters where you can spare or be friendly towards folks, and I shot first and asked questions later without any substantial changes to the dialog.
In some of the harder modes, going hostile in areas where you don't have to pick a fight can cause complications. For instance, there are some peaceful town areas that don't look heavily patrolled, but hard enemies and automatons will spawn if you cause trouble.
However, I've found I can hit and run and not really fight all these optional enemies if I choose not to. Unlike the scripted encounters where a locked door will prevent you from proceeding until combat is resolved, you can steal/murder/etc with impunity as long as you can run faster than they can. Eventually you'll trigger another scripted sequence and the enemies will despawn.
I'll also mention that I can't recall a time when I looted a non-hostile character's body and found anything - if you're low on ammo or looking for money, this is certainly not a reliable way to get more.
There is no downside to killing NPC's with the exception of children which you cannot kill. In some situations killing NPC's will render other characters in your area as hostile but other than that there is no change to the greater storyline.