Need answer fast?
I know it still wouldn’t work, but I think it would be better to knot the sheet under an armpit (or 2 corners under both armpits?) than under the crotch.
Related question: If you’re in a car or even house that is hurtling off a cliff, can you open the door and just step out a fraction of a second before impact?
First off, being on top of the door instead of trying to hang on to it from underneath is marginally more effective. Mythbusters demonstrated that the grip factor would make holding on impossible. However, you’d almost immediately get some gust or instability and flip, and thereby be hanging under the door instead of surfing.
Even assuming a magical day, and you somehow stayed on top all the way down, and the board stayed level and didn’t tip over, there’s still not enough surface area to do much. Your terminal velocity will be a bit lower than ~200 mph for nominal human, spread wide, but not slow enough to prevent severe damage on impact.
Now the question about bedsheets gets interesting if you propose that you have several sheets available. Could you jury rig a makeshift parachute by tying sheet corners together, then tying the end corners to a piece of rope, and, say, holding the rope under your arms? How many sheets would that take? 4 King size sheets? 12?
Yes, but it would still kill you. This would be similar to being in a falling elevator and jumping into the air just before it hits the ground. You could not shed enough of your acceleration.
The Mythbusters tested the elevator theory by dropping an elevator in an abonded building eight stories with a dummy inside. The dummy was rigged so that it would be launched in the air a fraction of a second before the elevator hit the ground. They determined the elevator fell at a rate of 53 mph. The dummy was launched into the air at 2 mph. The dummy hit the ground at an effective speed of 51 mph. The dummy was destroyed. They used instruments to determine that the acceleration force of the dummy would have been fatal to a human being.
If you had rope for lines, and could connect the sheet in between the corners also, you should be in pretty good shape. You could make something that has around the same area of common parachutes, and you have lines for more points of attachment and something better to hold you and the chute together. Unless you stitched the sheets together you’ll be venting more air through than a normal parachute, but it should be able to slow your descent sufficiently to live. That is if the sheets don’t tear apart. If you have enough rope you could stitch a net together that goes over the top of the chute to distribute your weight better. So if you have four king size bed sheets, 300 - 400 feet of rope, a small knife to cut rope and make holes for stitching, a lot of time to construct the parachute before you are burned alive or shot by your arch enemy, not too many pointy things on the ground, fairly calm air, and a lot of luck, then you could parachute off the building and land safely, or maybe with a couple of minor fractures. Definitely not a splat though. Of course if you have enough time to prepare, you could just go and buy a parachute and use that. Or just not jump off a tall building, which is the simplest solution.
ETA: previous post: You might get 30 sq.ft. of effective area from a king size bedsheet, so 4 sheets would only handle a person who weighed 120 pounds. Experienced base jumpers do it with less than 1 sq.ft. per pound, but you don’t have a real parachute and probably aren’t an experienced base jumper (and if you were you’d never go the top of a tall building without your chute), so 1 sq.ft. per pound is probably the limit before you die.
Yeah, if you have a lot of time to make an elaborate construction, you might have time to try other methods. That’s why I limited the construction to tying corners together. I figure rope from the four outer corners gives you the most useful surface area, as tying ends together both takes up material in the knots, and reduces effectiveness by bunching the sheet ends inward. Rope allows to let the sides out as much as possible. But constructing a net or a harness would be too labor intensive.
I was just trying to back of envelope get a scale of what it would take. So 6 king sheets might get somewhere close, but you’re relying on your knot skills and having some rope to allow as much spread as possible. Tying the outer corners to hold onto is going to significantly reduce effectiveness, but how much?
“Well, see, if you have 10 king size sheets and tie all the corners together…”
Um, yeah.
Y’know, if the first response to the OP had been “Yes” we wouldn’t be having this conversation.
You could still tie nots inbetween the corners to make a more effective chute. It will chew up surface area though. And just 4 ropes isn’t going to distribute the weight very well. You’d likely end up with the plummeting sack with only a few sq.ft. of effective surface area. The very early concepts of parachutes tried to account for this with a wooden frame, though I don’t know if there’s any conclusive evidence that any of those ideas were ever tried.
Also if you land on something soft, you could live. You may never walk again, or even regain conciousness though. But I knew a Green Beret who survived a malfunction. A lot of broken bones. But the guys who knew him figured he survived because he landed on his head 
Well, it works for Bugs Bunny.
That’s me, the fly in the ointment.