What a fascinating thing to contemplate. Yes, everything above will go outwards, but I am not sure that it will go off into deep space, not even if you cut it above center of the mass (geo orbit). Essentially you will have an object in orbit where you have removed some of the mass and shifted the center of mass considerably in the non-uniform gravity field, but the velocity is unchanged. This will result in a high elliptic orbit, if it is rigid. I don't think it will be rigid. So you will at least for some time have a large whip spinning and yanking, possibly ripping itself apart, in an highly irregular orbit. Which is fine, not much so far out anyway.
If you sever the bit near the Atmosphere I agree that it will fall down and the bit furthest out will burn up before it hits. Something I would love to see and nobody will be able to compute the outcome of. The wire in front should push the air aside, no? So higher speed, so you might get friction heating to air on the sides instead of compression in front. I dunno, as I said, would love to see it.
But what if you cut it further out, lets say 10 000 km up. It would fall down, but it's velocity and altitude of the center of mass would indicate that it should be able to enter an elliptic orbit. However it is tied to the ground, but that bit would burn off some distance into the atmosphere. Which means the burnt off end would dangle into the atmosphere and drag the rest down unless it gets yanked out by the rest of the wire gaining higher speed. Which it would, but then it would probably bounce back and frequently touch the atmosphere. So for days or months the remaining wire would whip one of the ends into the atmosphere burning it off in what I will hope be a spectacular manner.