So I just leaned over the desk to check the hookups on my tower, and I got a whiff of sweet-smelling vapor that seems to emit from the top-back vents of the monitor. What is this? Is it bad to smell?
Slightly ionized air can have that “fresh after rainfall” smell. Not harmful in small quantities, but ozone can be irritating in larger amounts. Can be a side effect of running electrical equipment.
Unless I’m totally off-base here.
Does it have, like, a “new monitor smell?” I notice that a lot of plastics or composites can throw off a sweet smell when heated but not combusted.
Yeah, how old is the monitor?
If it’s brand new it’s quite common to smell the wiring insulation and other plastics as it heats up for the first few times.
If it’s a new smell in an old monitor I’d be a lot more suspicious.
Not certain, certainly, but I am inclined to guess that you’re smelling ozone, which is a molecular form of oxygen having three atoms instead of the more usual two. It is generated from atmospheric oxygen by electrical and ionizing discharges, and it often occurs with ionized air, though the two are distinctly different things.
I would guess it is generated by discharge of high voltage on some tiny or sharp bit of wire at the end of a fluorescent tube (backlight) in your monitor.
Ozone is a respiratory irritant and it helps degrade some elastomers (they get dusty and dry and crumbly on the surface. It is also used to purify drinking water (it’s like chlorine in that regard, but safer in several ways).
If it’s ozone you’re smelling, and you ae getting too much, you might notice symptoms of short-term irritation similar to allergy or to working with ammonia or chlorine bleach. Otherwise, no problem.
The monitor is a couple of years old. And, I’m glad that this seems to be no more than an “irritant”; thank you. 
Everyone’s monitor smells good to himself.