A cone standing on its tip is considered to be in unstable equilibrium as a slightest force could topple it. So, if the cone is stood on its tip with no other force other than gravity (and the corresponding ground reaction force), will it continue to stand without toppling? Has this been attempted experimentally?
If you made the most perfect cone possible, so that its tip was a single atom, and stood it on the most perfect surface possible (a perfectly smooth, perfectly hard sheet of atoms), and completely removed all forces other than gravity, it would still topple. This is because those atoms are all jiggling around due to thermal motion. This effect fundamentally cannot be removed, and sooner or later it will cause enough of a perturbation to cause the cone to start falling. It might take a fairly long time before this happens, however.
A more realistically imperfect cone might have a better chance. A real metal object isn't perfectly smooth. It's fairly likely that the tip of the cone will actually be rounded, even if it appears perfectly sharp under a microscope, and on an even smaller level it will be quite jagged. The same applies to the surface. It's possible that the two will find a way to mesh together such that they balance, especially since neither will really be perfectly hard, so they'll inevitably sink into one another a bit.
As for whether the experiment has been done, I'm sure a lot of people have made cones of varying degrees of imperfection and attempted to balance them with varying degrees of success. But the idealised experiment - perfect cone, perfect surface, no other forces - would be rather expensive. Creating and checking the cone tip and the surface would require an electron microscope, but the really hard part is eliminating all other forces. Air movement, seismic vibrations, electromagnetic effects, perhaps even the gravitational field of nearby objects would all have to be accounted for. For these reasons I doubt that version of the experiment has been done.
This is going beyond what's practically possible, but in principle, thermal effects could be eliminated (or very nearly eliminated) by cooling the system down to absolute zero. It's impossible to completely reach absolute zero, but we can get very close. But if we do that we run into a different problem: Heisenberg's uncertainty principle. The problem is that in order for the cone to be perfectly balanced, its centre of mass must be exactly above the point of the cone, and also the cone has to not be moving at all, i.e. it must have exactly zero momentum. But if the cone was set up in that way we would know the exact position of the centre of mass, and we'd also know its exact momentum at the same time. Quantum mechanics says this is impossible, and this means that even if you could eliminate thermal effects completely, it would be fundamentally impossible to set the cone up so that it was absolutely, perfectly balanced.
A few seconds. What "a few" is numerically depends on the author of such calcuations, but generally something like 4 or 5 seconds, maybe up to 10. Quantum mechanically, it must fall. Which direction, no one can say. The perfect symmetry of an ideal cone on an ideal flat surface will be broken as the object "chooses" some direction. For realistic materials, microscopic imperfections, thermal motion of atoms, and electromagnetic fluctuations will surely perturb the object to fall in a particular direction.
There was an article in American Journal of Physics, probably early 1970s, that either explained the calculation for a pencil standing on its tip, or made reference to another paper that did the calculation. I looked, but did not find it right now. But others have written about this topic.
Try this more recent paper: http://rickbradford.co.uk/HowLongCanAPencilRemainBalancedOnItsTip.pdf or http://mccammon.ucsd.edu/~jcsung/qmp.pdf
Whatever the proportions of your cone, if it's a cone, pencil shape, or a pin, the difference won't matter much. The time it stands may vary some due to different moment of inertia, but nothing wild.
If attempted experimentally, it is possible for an object to stand on point slightly longer. As it starts to fall in one direction, a tiny random jiggle of the table due to sound waves, a truck going by outside the building etc, may happen to move the point back under the center of mass of the object, delaying its fall. But I won't be placing any bets on the object standing for, say, half a minute.
If simulated in a computer, using classical mechanics and having the pointy object placed exactly at the origin and using a perfectly symmetric grid, it could stay up forever. This is a pathological result, however. Simulations that are too perfect have trouble breaking symmetry.
Two additions to the other good answers: