Is it normal for the furnace flue to be wrapped in insulation that looks blackened?

Wrapping a furnace flue in insulation is not a thing, at least not on any gas furnace built in the last several decades. That pipe is almost certainly Type B vent, wich is the double-wall stuff designed to let air move between the layers and keep the outer surface cool. Covering it in raw fiberglass does the opposite, it holds heat against the pipe instead of letting it shed, so whoever did it either didn't know what they were looking at or just wanted to pack the closet for appearance's sake. I'd say the duct nearby probaly did need insulating and they got lazy and did both. The dark crumbly part is most likely dust that's been baked onto the insulation over years of heat cycles, which looks grimy but is harmless. Easy way to check: wipe it with a dark paper towel or cloth. If it comes away as dry gray dust, you're fine. If it smears black or feels greasy, that's soot, and soot means the burn isn't clean or there's a leak at a joint.

Peeling the insulation back to look is a perfectly reasonable DIY move. Don't disconnect anything, just pull the batting away and look at the pipe itself. You want to see uniform metal, no rust streaks, no black smudges above or below the joints, no chalky white residue. Discoloration like heat scorching or blueing on the pipe is a sign it's running hotter than it should.

One thing worth checking while you're in there: if the pipe is actually single-wall galvanized running through a closed closet, that's a diffrent situation entirely and deserves an HVAC look. And since you're renting, photograph whatever you find and send it to the landlord. If it turns out to be soot, the service call is on them, not you.

Beyond the visual issue with the insulation, that trapped heat is an efficiency red flag. If the system is fighting to dump heat into a cramped, insulated space, the combustion process itself is probably compromised. On a similar job last year, I found a vent pipe running at 450 degrees F when the proper operating temperature should be under 250 degrees F. That kind of excess heat often signals a problem with the burner tuning or the heat exchanger integrity. Make sure you look closely at the outlet point where the pipe enters the chimney or wall.

I've dealt with enough of these setups over the years that I know what the dark marks are. If you're seeing heat stress, that insulation trap is the problem, period. Especially if that pipe is galvanized, which they used a long time ago before all the newer PVC stuff. If it's hot enough to bake that fiberglass, that metal is under strain. I always tell folks that if you see any signs of corrosion or pitting on an old vent, you don't try to patch it up. You call in a pro to swap the whole line. Those older joints are weak once the heat cycles start working on them.

I totally agree with the part about the insulation on the vent pipe being a bad move. Most folks just don't know that those vent pipes are designed to dissipate heat, not trap it. I had a friend who wrapped a dryer vent with fiberglass years ago, and the whole thing started smelling like burning plastic within a year. The trapped heat warped the plastic ducting and the metal crimps on the end started seizing up from the constant high temp.

If that pipe in your closet is running significantly hotter than it should, it's basicly cooked itself from the inside out, so letting it sit there while baking is not good. You definately need to get that stuff pulled back to get a proper look.