Couple options...
1) Take I-35 to Duluth, then take highway 61 (the "scenic" North Shore Drive) northbound. You can turn north on Route 2 in Two Harbors and after about 15 miles, there will be some National Forest roads on the left. Also some obvious logging roads.
2) If you follow Route 2 to where it T's on Rt 15 (about 25 miles north of Two Harbors), take a right on 15, go less than a mile, and you'll find the Stony Grade Road, which is part of the Finland State Forest. It heads north to an area called Scott Junction (just a stretch of road). More remote country back in there. Slightly OT, but that's also a good place to spot a Canada lynx, if you're lucky (big snowshoe hare population in there, which is their main food source).
3) Or...if you go farther up 61, you can take Route 1 north through Finland towards Isabella. Outside of Finland you start seeing the same type of roads. After Isabella, though, you're really in the boonies. LOTS of unmarked "roads to nowhere." A lot are private property and marked as such, but many are not. If you turn directly into the National Forest when you hit the 90-degree left turn in Isabella, you're on forest service roads from there on and the sky's the limit. No houses, some cool lakes, etc.
4) I forget whether it's before the Stony Grade road or after it, but Route 2 continues North after the 'T' north of Two Harbors (though it will be a distinct left turn) and joins up with Route 1 North of Isabella. There's a whole lot of nothing in that stretch. Good moose-spotting area, too, as is highway 1 north of Finland.
5) Another way to go from Duluth is to take 53 north towards Virginia, then get on 169 towards Ely just outside Virginia. As you get closer to Ely, there some more forest roads, but a LOT more after it (not well-traveled up there). Some aren't even marked. For some reason, I always end up on these (?).
6) 53 N towards I-Falls also has some remote roads, but that's pretty far for a day trip (this is where that whole "expedition vehicle" concept would come in handy!)
Lots of general poking-around roads around Cook, Babbit, Embarass, etc.
7) These are all a pretty good distance from the cities. A closer alternative is the Nemadji State Forest, which is to the east off of I-35, between Hinckley and Duluth. Not as remote as the other stuff I mentioned, but better than trying to off-road in Bloomington (!).
If you have time and/or go through Duluth on a weekday (like a Friday), a couple of good places to stop would be at the Superior National Forest HQ on Grand Avenue in Duluth and the DNR Forestry Office in Two Harbors (right on Route 2, north of town). Stop in and pick up some maps.
More when I can think of it. Hopefully these maps will give some sense to what I'm saying.
Ron