ATV's, including UTV's and (IIRC) dirt bikes are not allowed on the White Rim Road (or anywhere else inside the National Parks down there)... last I checked. 4x4's, Jeeps, Etc, are.
I drove the road in 2.5 days... I had to cut my trip a day short when my 5gal water jug sprung a leak from bouncing/sliding in the bed of my pickup. With only a few quarts of water didn't seem like a wise way to spend the next 36 hours on the trail. I was alone, and it was in February. I had the means, and a plan to hike out, if required, but I didn't risk it. I didn't see a soul once I hit dirt. It was wonderful.
In the later winter and early spring, rockfall can block the trail, or portions of it, requiring a long return trip back the way you came. I had to drive right on the edge of the trail, in a steep cliffed up section to get around a boulder that had fallen down and blocked most of the road. When I went back and made the rangers aware of the boulder, they dispatched a man in a jeep with some explosives to remove it.
The Green River side can have some pretty serious sandy sections once the road drops down to the river. Some areas can be very muddy and slick as well. The trail itself is not overly technical, and there are not any real major obstacles. These conditions change constantly, and one good storm can wipe out a road down there in minutes.
Take extra water and fuel. It's only about 100 miles of trail, but the slow going eats fuel, as do any of the short side roads you might take. I needed to top off my fuel tank from my jerry can to make it back to Moab.
The views are amazing. I could have spent a week exploring the area, and still felt like I was rushing.
You'll need a backcountry permit to camp overnight, and during some parts of the year, all the camp sites are reserved months ahead of time. It can be driven in a day... but its a VERY long day, and you'll miss a lot going so fast.